the invention will now be described with reference to fig1 and 1a , which illustrate a side elevation view in cross section of the midsole 103 . the outsole 105 is not part of the midsole 103 . a sockliner 101 is not part of the midsole 103 . the midsole 103 is shown beneath the sockliner 101 . the outsole 105 of the shoe is beneath the midsole 103 . the dual density midsole is located between the shoe upper ( not shown ) and the outsole 105 . the midsole 103 , as shown in fig1 a , comprises an upper layer 107 and a lower layer 109 . the upper layer 107 and / or the lower layer 109 may themselves each be comprised of two or more sub - layers . the upper layer 107 has a top surface 113 substantially opposite a bottom surface 115 . the lower layer 109 has a top surface 117 substantially opposite a bottom surface 121 . the shoe has a front tip 140 located at the farthest point toward the front of the shoe and a rear tip 142 located at the farthest point toward the rear of the shoe . the upper layer 107 includes a toe region 151 that extends substantially from the medial side of the shoe to the lateral side of the shoe at a location that begins in the vicinity of the front tip 140 and extends from there to a location that is approximately one third of the distance toward the rear tip 142 . the lower layer 109 includes a toe region 161 that extends substantially from the medial side of the shoe to the lateral side of the shoe at a location that begins in the vicinity of the front tip 140 and extends from there to a location that is approximately one third of the distance toward the rear tip 142 . the outsole 105 includes a toe region 171 that extends substantially from the medial side of the shoe to the lateral side of the shoe at a location that begins in the vicinity of the front tip 140 and extends from there to a location that is approximately one third of the distance toward the rear tip 142 . the upper layer 107 includes a heel region 153 that extends substantially from the medial side of the shoe to the lateral side of the shoe at a location that begins in the vicinity of the rear tip 142 and extends from there to a location that is approximately one third of the distance toward the front tip 142 . the lower layer 109 includes a heel region 163 that extends substantially from the medial side of the shoe to the lateral side of the shoe at a location that begins in the vicinity of the rear tip 142 and extends from there to a location that is approximately one third of the distance toward the front tip 140 . the outsole 105 includes a heel region 173 that extends substantially from the medial side of the shoe to the lateral side of the shoe at a location that begins in the vicinity of the rear tip 142 and extends from there to a location that is approximately one third of the distance toward the front tip 140 . the upper layer 107 includes a middle region 152 that extends substantially from the medial side of the shoe to the lateral side of the shoe at a location that extends approximately between the toe region 151 and the heel region 153 . the lower layer 109 includes a middle region 162 that extends substantially from the medial side of the shoe to the lateral side of the shoe at a location that extends approximately between the toe region 161 and the heel region 163 . the outsole 105 includes a middle region 172 that extends substantially from the medial side of the shoe at a location that extends approximately between the toe region 171 and the heel region 173 . typically , the lower layer 109 is on average thicker in the heel region 163 than it is in the toe region 161 . typically , the thickness of the lower layer 109 is less than about 45 millimeters in the heel region 163 and has an average thickness in the heel region 163 of at least about 6 . 5 millimeters , and is less than about 25 millimeters in the middle region 162 and the toe region 161 and has an average thickness in the middle region 162 and the toe region 161 of at least 3 millimeters . the upper layer 107 has a first density and the lower layer 109 has a second density that is different from the first density and is typically less dense than the first density . the upper layer 107 has a first compressibility and the lower layer 109 has a second compressibility that is different from the first compressibility . the compressibility of the lower layer 109 is typically relatively high . due to this relatively high compressibility , the lower layer 109 undergoes a relatively high amount of deformation when subjected to a given load . the upper layer 107 is typically made from polyurethane , polyvinyl chloride , rubber or thermal plastic rubber . however , the upper layer 107 can be made from any other material without departing from the scope of the present invention . typically the upper layer 107 will have a density of between about 0 . 400 and about 0 . 500 grams per cubic centimeter and a durometer between about 50 and about 75 shore a ( astm d2240 ). the lower layer 109 is made of a compressible and deformable yet resilient material which may or may not be the same material of which the upper layer 107 is made . typically the lower layer 109 will have a density of between about 0 . 325 and about 0 . 419 grams per cubic centimeter and a durometer between about 15 and about 38 shore a ( astm d2240 ). the upper layer 107 has a top surface 113 that is typically positioned below an insole board ( not shown ) which is typically positioned below the sockliner 101 . the upper layer 107 also has a bottom surface 115 that is secured to and in substantially continuous contact with the top surface 117 of the lower layer 109 by either friction and / or an adhesive and / or other similar means . alternatively , substantially the entire bottom surface 115 of the upper layer 107 may be molded to substantially the entire top surface 117 of the lower layer 109 . the outsole 105 has a top surface 119 . the bottom surface 121 of the lower layer 109 is positioned above the top surface 119 of outsole 105 . when viewed while moving from the frontmost point 150 of the upper layer 107 to the rearmost point 154 of the upper layer 107 , the bottom surface 115 of the upper layer 107 , as shown in a preferred embodiment in fig1 a , has a convexity 180 that comprises at least a downward curve 190 located in at least a portion of the toe region 151 . all convexities identified by an element number in this specification are convexities that , to some degree , protrude from , and are part of , their respective bottom surface 115 , 315 , or 815 of the respective upper layer 107 , 307 or 807 . downward curve , as used here and throughout this specification , unless otherwise noted , refers to a direction that moves toward the ground from any specified location on the shoe when viewed while moving from a front tip 142 , 342 , or 842 to a respective rear tip 140 , 340 , or 840 and while the shoe is oriented in its typical upright position where a bottom surface 123 , 323 or 823 of the respective outsole 105 , 305 or 805 is in unloaded contact with the ground . the downward curve 190 of convexity 180 begins at , or near the vicinity of , the frontmost point 150 of the upper layer 107 and gradually and continuously descends downwardly from there through at least a portion of the toe region 151 . the portion of the upper layer 107 indicated by lines extending from , and associated with , element number 180 indicates the approximate range wherein convexity 180 is typically primarily located . convexity 180 may , or may not , be entirely located within the range indicated by the lines extending from , and associated with , element number 180 . convexity 180 , as shown in a preferred embodiment in fig1 a , is relatively shallow due to its large radius , or radii , of curvature . convexity 180 may comprise a curve or curves in addition to downward curve 190 . the radius of curvature throughout convexity 180 may be completely constant , may have one or more constant portions mixed with one or more non - constant portions , or may be completely non - constant . downward curve 190 , as well as any other curve or curves that are part of convexity 180 , may , at any point on any of those curves , have a slope somewhere between negative infinity and positive infinity and can include a slope that is zero , gradual , moderate , steep , vertical or somewhere between any of those amounts . although the downward curve 190 of convexity 180 is shown in fig1 a as beginning near the frontmost point 150 , downward curve 190 of convexity 180 may instead begin at some other location on the upper layer 107 . only a portion of convexity 180 may be located in the toe region 151 . alternatively , all or substantially all of convexity 180 may be located in the toe region 151 . convexity 180 , or a portion thereof , may occupy all of the toe region 151 . alternatively , convexity 180 , or a portion thereof , may occupy a substantial portion of the toe region 151 . convexity 180 has a first wavelength and a first amplitude . the bottom surface 115 of the upper layer 107 , as shown in fig1 a , has a convexity 181 that comprises at least a downward curve 191 located in at least a portion of the middle region 152 . in this preferred embodiment , convexity 181 further comprises at least an upward curve 192 . upward curve , as used here and throughout this specification , unless otherwise noted , refers to a direction that moves away from the ground from any specified location on the shoe when viewed while moving from a front tip 142 , 342 , or 842 to a respective rear tip 140 , 340 , or 840 and while the shoe is oriented in its typical upright position where a bottom surface 123 , 323 or 823 of the outsole 105 , 305 or 805 is in unloaded contact with the ground . downward curve 191 may or may not be contiguous with upward curve 192 . downward curve 191 descends downwardly in at least a portion of the middle region 152 . upward curve 192 ascends upwardly in at least a portion of the middle region 152 . the portion of the upper layer 107 indicated by lines extending from , and associated with , element number 181 indicates the approximate range wherein convexity 181 is typically primarily located . convexity 181 may , or may not , be entirely located within the range indicated by the lines extending from , and associated with , element number 181 . convexity 181 has a relatively pronounced bulge due to its relatively small radius , or radii , of curvature . convexity 181 may comprise a curve or curves in addition to downward curve 191 and upward curve 192 . the radius of curvature throughout convexity 181 may be completely constant , may have one or more constant portions mixed with one or more non - constant portions , or may be completely non - constant . downward curve 191 , upward curve 192 , as well as any other curve or curves that are part of convexity 181 , may , at any point on any of those curves , have a slope somewhere between negative infinity and positive infinity and can include a slope that is zero , gradual , moderate , steep , vertical or somewhere between any of those amounts . although the downward curve 191 of convexity 181 is shown in fig1 a as beginning near the middle of the middle region 152 and ending at a location closer to the heel region 153 than the middle of the middle region 152 , downward curve 191 of convexity 181 may instead begin at some other location on the upper layer 107 and end at some other location on the upper layer 107 . although the upward curve 192 of convexity 181 is shown in fig1 a as beginning near the middle of the middle region 152 and ending in the middle region at a location near the heel region 153 , upward curve 192 of convexity 181 may instead begin at some other location on the upper layer 107 and end at some other location on the upper layer 107 . only a portion of convexity 181 may be located in the middle region 152 . alternatively , all or substantially all of convexity 181 may be located in the middle region 152 . convexity 181 , or a portion thereof , may occupy all of the middle region 152 . alternatively , convexity 181 , or a portion thereof , may occupy a substantial portion of the middle region 152 . convexity 181 has a second wavelength that is typically different from the first wavelength of convexity 180 . convexity 181 has a second amplitude that is typically different from the first amplitude of convexity 180 . line 2 - 2 is at or near the lowest point of convexity 181 . the primary purpose of convexity 181 is to reduce — but not eliminate — compression and deformity of the lower layer 109 in the region of the convexity 181 and to provide stability . fig2 shows how convexity 181 extends substantially from the lateral to medial side of the upper layer 107 . convexity 180 may or may not be contiguous with convexity 181 . the bottom surface 115 of the upper layer 107 , as shown in fig1 a , has a concavity 182 that comprises at least an upward curve 193 located in at least a portion of the heel region 153 . all concavities identified by an element number in this specification are concavities that , to some degree , form a depression in , and are part of , the respective bottom surface 115 , 315 , or 815 of the respective upper layer 107 , 307 or 807 . in this preferred embodiment , concavity 182 further comprises at least a downward curve 194 . upward curve 193 may or may not be contiguous with downward curve 194 . upward curve 193 ascends upwardly in at least a portion of the heel region 153 . downward curve 194 descends downwardly in at least a portion of the heel region 153 . the portion of the upper layer 107 indicated by lines extending from , and associated with , element number 182 indicates the approximate range wherein concavity 182 is typically primarily located . concavity 182 may , or may not , be entirely located within the range indicated by the lines extending from , and associated with , element number 182 . concavity 182 has a relatively moderate depression due to its relatively moderate radius , or radii , of curvature . concavity 182 may comprise a curve or curves in addition to upward curve 193 and downward curve 194 . the radius of curvature throughout concavity 182 may be completely constant , may have one or more constant portions mixed with one or more non - constant portions , or may be completely non - constant . upward curve 193 , downward curve 194 , as well as any other curve or curves that are part of concavity 182 , may , at any point on any of those curves , have a slope somewhere between negative infinity and positive infinity and can include a slope that is zero , gradual , moderate , steep , vertical or somewhere between any of those amounts . although the upward curve 193 of concavity 182 is shown in fig1 a as beginning at a location where the heel region 153 and the middle region 152 transition into one another , the upward curve 193 of concavity 182 could instead begin at some other location on the upper layer 107 . although the upward curve 193 of concavity 182 is shown in fig1 a as ending at a location near the middle of the heel region 153 , upward curve 193 may instead end at some other location on the upper layer 107 . although the downward curve 194 of concavity 182 is shown in fig1 a as beginning near the middle of the heel region 153 and ending in the vicinity of the rearmost point 154 of the upper layer 107 , downward curve 194 of concavity 182 may instead begin at some other location on the upper layer 107 and end at some other location on the upper layer 107 . convexity 181 may or may not be contiguous with concavity 182 . only a portion of concavity 182 may be located in the heel region 153 . alternatively , all or substantially all of concavity 182 may be located in the heel region 153 . concavity 182 , or a portion thereof , may occupy all of the heel region 153 . alternatively , concavity 182 , or a portion thereof , may occupy a substantial portion of the heel region 153 . concavity 182 has a third wavelength that is typically different from both the first wavelength of convexity 180 and the second wavelength of convexity 181 . concavity 182 has a third amplitude that is typically different from both the first amplitude of convexity 180 and the second amplitude of convexity 181 . in preferred embodiments , the top surface 117 of the lower layer 109 of the midsole 103 is in substantially continuous contact with the bottom surface 115 of the upper layer 107 of the midsole . due to this substantially continuous contact between top surface 117 and bottom surface 115 in these preferred embodiments , each convexity in the bottom surface 115 has a corresponding concavity in the top surface 117 and each concavity in the bottom surface 115 has a corresponding convexity in the top surface 117 . in other embodiments , such substantially continuous contact between top surface 117 and bottom surface 115 may not be present . the outsole 105 has a top surface 121 and a bottom surface 123 . the outsole 105 may curve upwardly in the heel region . when the shoe is in its typical upright , unloaded state , the frontmost point 170 is relatively high above the ground . from a point at or near the vicinity of the frontmost point 170 , the outsole 105 has a gradual downward curve 195 that continues through at least a portion of the toe region 171 of the outsole 105 until it becomes straight or nearly straight at some point in the middle region 172 of the outsole 105 . starting in this middle region 172 , the outsole 105 has a gradual , upward curve 196 that continues to curve upward through at least a portion of the heel region 173 of the outsole 105 . this gradual upward curve 196 typically continues until the outsole 105 approaches the vicinity of the rear tip 142 of the shoe . this upward curve 196 is typically sharper than downward curve 195 in the toe region 171 . upward curve 196 may be substantially sharper than shown in fig1 a or substantially shallower than shown in fig1 a . the bottom surface 123 of the outsole 105 typically contains grooves and / or patterns for optimal traction and wear . fig2 illustrates a front elevation view in cross section of fig1 along line 2 - 2 in the direction of the appended arrows . fig2 shows the construction and placement of the upper layer 107 on top of the lower layer 109 with the convexity 181 sitting in the congruent curved recess or depression 111 . the cross sectional shape of the bottom surface 115 of the upper layer 107 and the top surface 117 of the lower layer 109 at line 2 - 2 is shown in fig2 as a single line that is horizontal at one end , then dips downwardly toward the middle , is horizontal in the middle , then slopes upwardly at the other end and is horizontal at the other end . the invention will now be described with reference to a preferred embodiment shown in fig3 and 3a . this embodiment shows a side elevation view in cross section of the midsole 303 and the outsole 305 of the shoe . the midsole 303 , as shown , comprises two layers . typically , the lower layer 309 of the midsole 303 is on average thicker in the heel region 363 of the shoe than it is in the toe region 361 . typically , the thickness of the lower layer 309 is less than about 45 millimeters thick in the heel region 363 of the shoe and has an average thickness in the heel region 363 of at least about 6 . 5 millimeters , and is less than about 25 millimeters thick in the middle region 362 and the toe region 361 of the shoe and has an average thickness in the middle region 362 and the toe region 361 of at least about 3 millimeters . the upper layer 307 has a first density and the lower layer 309 has a second density different from the first density and is typically less dense than the first density . the upper layer 307 has a first compressibility and the lower layer 309 has a second compressibility that is different from the first compressibility . the compressibility of the lower layer 309 is typically relatively high . due to this relatively high compressibility , the lower layer 309 undergoes a relatively high amount of deformation when subjected to a given load . the upper layer 307 is typically made from polyurethane , polyvinyl chloride , rubber or thermal plastic rubber . however , the upper layer 307 can be made from any other material without departing from the scope of the present invention . typically the upper layer 307 will have a density of between about 0 . 400 and about 0 . 500 grams per cubic centimeter and a durometer between about 50 and about 75 shore a ( astm d2240 ). the lower layer 309 is made of a compressible and deformable yet resilient material which may or may not be the same material of which the upper layer 307 is made . typically the lower layer 309 will have a density of between about 0 . 325 and about 0 . 419 grams per cubic centimeter and a durometer between about 15 and about 38 shore a ( astm d2240 ). the top surface 313 of the upper layer 307 is typically positioned below an insole board ( not shown ) which is typically positioned below the sockliner 301 . the upper layer 307 has a bottom surface 315 that is located above the top surface 317 of the lower layer 309 . the lower layer 309 has a bottom surface 321 . the outsole 305 has a top surface 319 . the bottom surface 321 of the lower layer 309 is located above the top surface 319 of the outsole 305 the bottom surface 315 of the upper layer 307 , as shown in a preferred embodiment in fig3 a , has a convexity 380 that comprises at least a downward curve 390 located in at least a portion of the toe region 351 . the downward curve 390 of convexity 380 begins at , or near the vicinity of , the frontmost point 350 of the upper layer 307 and gradually and continuously descends downwardly from there through at least a portion of the toe region 351 . the portion of the upper layer 307 indicated by lines extending from , and associated with , element number 380 indicates the approximate range wherein convexity 380 is typically primarily located . convexity 380 may , or may not , be entirely located within the range indicated by the lines extending from , and associated with , element number 380 . convexity 380 , as shown in a preferred embodiment in fig3 a , is relatively shallow due to its large radius , or radii , of curvature . convexity 380 may comprise a curve or curves in addition to downward curve 390 . the radius of curvature throughout convexity 380 may be completely constant , may have one or more constant portions mixed with one or more non - constant portions , or may be completely non - constant . downward curve 390 , as well as any other curve or curves that are part of convexity 380 , may , at any point on any of those curves , have a slope somewhere between negative infinity and positive infinity and can include a slope that is zero , gradual , moderate , steep , vertical or somewhere between any of those amounts . although the downward curve 390 of convexity 380 is shown in fig3 a as beginning near the frontmost point 350 , downward curve 390 of convexity 380 may instead begin at some other location on the upper layer 307 . although convexity 380 is shown in fig3 a as ending at a location in the middle region 352 or the location where the middle region 352 transitions into the heel region 353 , convexity 380 may end at some other location on the upper layer 307 . the bottom surface 315 of the upper layer 307 , as shown in fig3 a , has a concavity 382 that comprises at least an upward curve 393 located in at least a portion of the heel region 353 . in this preferred embodiment , concavity 382 further comprises at least a downward curve 394 . upward curve 393 may or may not be contiguous with downward curve 394 . upward curve 393 ascends upwardly in at least a portion of the heel region 353 . downward curve 394 descends downwardly in at least a portion of the heel region 353 . the portion of the upper layer 307 indicated by lines extending from , and associated with , element number 382 indicates the approximate range wherein concavity 382 is typically primarily located . concavity 382 may , or may not , be entirely located within the range indicated by the lines extending from , and associated with , element number 382 . concavity 382 has a relatively moderate depression due to its relatively moderate radius , or radii , of curvature . concavity 382 may comprise a curve or curves in addition to upward curve 393 and downward curve 394 . the radius of curvature throughout concavity 382 may be completely constant , may have one or more constant portions mixed with one or more non - constant portions , or may be completely non - constant . upward curve 393 , downward curve 394 , as well as any other curve or curves that are part of concavity 382 , may , at any point on any of those curves , have a slope somewhere between negative infinity and positive infinity and can include a slope that is zero , gradual , moderate , steep , vertical or somewhere between any of those amounts . although the upward curve 393 of concavity 382 is shown in fig3 a as beginning at a location where the middle region 352 and the heel region 353 transition into one another , the upward curve 393 of concavity 382 could instead begin at some other location on the upper layer 307 . although the upward curve 393 of concavity 382 is shown in fig3 a as ending at a location near the transition between the middle region 352 and the heel region 353 , upward curve 393 may instead end at some other location on the upper layer 307 . although the downward curve 394 of concavity 382 is shown in fig3 a as beginning at a location near the transition between the middle region 352 and the heel region 353 and ending in the vicinity of the rearmost point 354 of the upper layer 307 , downward curve 394 of concavity 382 may instead begin at some other location on the upper layer 307 and end at some other location on the upper layer 307 . convexity 380 may or may not be contiguous with concavity 382 . the outsole 305 has a top surface 319 and a bottom surface 323 . the outsole 305 may curve upwardly in the heel region . when the shoe is in its typical upright , unloaded state , the frontmost point 370 is relatively high above the ground . from a point at or near the vicinity of the frontmost point 370 , the outsole 305 has a gradual downward curve 395 that continues through at least a portion of the toe region 371 of the outsole 305 until it reaches a virtually flat surface in the middle region 372 of the outsole 305 . starting in this middle region 372 , the outsole 305 has a gradual , upward curve 396 that continues to curve upward through at least a portion of the heel region 373 of the outsole 305 . this gradual upward curve 396 typically continues until the outsole 305 approaches the vicinity of the rear tip 342 of the shoe . this upward curve 396 is typically sharper than the curve in the toe region 371 . upward curve 396 may be substantially sharper than shown in fig3 a or substantially shallower than shown in fig3 a . the bottom surface 323 of the outsole 305 typically contains grooves and / or patterns for optimal traction and wear . fig4 shows a front elevation view in cross section of the midsole 303 shown in fig3 along line 4 - 4 in the direction of the appended arrows . as shown in fig4 , the bottom surface 315 of the upper layer 307 is in substantially continuous contact with the top surface 317 of the lower layer 309 . the cross sectional shape of the bottom surface 315 and the top surface 317 at line 4 - 4 is shown in fig4 by a substantially horizontal line that extends from the lateral side of the midsole 303 to the medial side . fig5 shows a front elevation view in cross section of the midsole 303 shown in fig3 along line 5 - 5 in the direction of the appended arrows . as shown in fig5 , the bottom surface 315 of the upper layer 307 is in substantially continuous contact with the top surface 317 of the lower layer 309 . the cross sectional shape of the bottom surface 315 and the top surface 317 at line 5 - 5 is shown in fig5 by a substantially horizontal line that extends from the lateral side of the midsole 303 to the medial side . fig6 shows a front elevation view in cross section of the midsole 303 shown in fig3 along line 6 - 6 in the direction of the appended arrows . as shown in fig6 , the bottom surface 315 of the upper layer 307 is in substantially continuous contact with the top surface 317 of the lower layer 309 . the cross sectional shape of the bottom surface 315 and the top surface 317 at line 6 - 6 is shown in fig6 by a substantially horizontal line that extends from the lateral side of the midsole 303 to the medial side . fig7 shows a front elevation view in cross section of the midsole 303 shown in fig3 along line 7 - 7 in the direction of the appended arrows . as shown in fig7 , the bottom surface 315 of the upper layer 307 is in substantially continuous contact with the top surface 317 of the lower layer 309 . the cross sectional shape of the bottom surface 315 and the top surface 317 at line 7 - 7 is shown in fig7 by a substantially horizontal line that extends from the lateral side of the midsole 303 to the medial side . as shown in fig4 - 7 , the cross sectional of the midsole 303 is of varying thickness , with there generally being a progression in thickness as the midsole 303 moves from the toe region to the heel region . in preferred embodiments , the top surface 317 of the lower layer 309 of the midsole 303 is in substantially continuous contact with the bottom surface 315 of the upper layer 307 of the midsole . due to this substantially continuous contact between top surface 317 and bottom surface 315 in these preferred embodiments , each convexity in the bottom surface 315 has a corresponding concavity in the top surface 317 and each concavity in the bottom surface 315 has a corresponding convexity in the top surface 317 . in other embodiments , such substantially continuous contact between top surface 317 and bottom surface 315 may not be present . the invention will now be described with reference to an alternative embodiment shown in fig8 and 8a . this embodiment shows a side elevation view in cross section of the midsole 803 and the outsole 805 of the shoe . the midsole 803 , as shown , comprises two layers . typically , the lower layer 809 of the midsole is on average thicker in the heel region 863 of the shoe than it is in the toe region 861 . typically , the thickness of the lower layer 809 is less than about 45 millimeters thick in the heel region 863 of the shoe and has an average thickness in the heel region 863 of at least about 6 . 5 millimeters , and is less than about 25 millimeters thick in the middle region 862 and the toe region 861 of the shoe and has an average thickness in the middle region 862 and the toe region 861 of at least about 3 millimeters . the upper layer 807 has a first density and the lower layer 809 has a second density different from the first density and is typically less dense than the first density . the upper layer 807 has a first compressibility and the lower layer 809 has a second compressibility that is different from the first compressibility . the compressibility of the lower layer 809 is typically relatively high . due to this relatively high compressibility , the lower layer 809 undergoes a relatively high amount of deformation when subjected to a given load . the upper layer 807 is typically made from polyurethane , polyvinyl chloride , rubber or thermal plastic rubber . however , the upper layer 807 can be made from any other material without departing from the scope of the present invention . typically the upper layer 807 will have a density of between about 0 . 400 and about 0 . 500 grams per cubic centimeter and a durometer between about 50 and about 75 shore a ( astm d2240 ). the lower layer 809 is made of a compressible and deformable yet resilient material which may or may not be the same material of which the upper layer 807 is made . typically the lower layer 809 will have a density of between about 0 . 325 and about 0 . 419 grams per cubic centimeter and a durometer between about 15 and about 38 shore a ( astm d2240 ). the top surface 813 of the upper layer 807 is typically positioned below an insole board ( not shown ) which is typically positioned below the sockliner 801 . the upper layer 807 has a bottom surface 815 that is located above the top surface 817 of the lower layer 809 . the lower layer 809 has a bottom surface 821 . the outsole 805 has a top surface 819 . the bottom surface 821 of the lower layer 809 is located above the top surface 819 of the outsole 805 . the bottom surface 815 of the upper layer 807 , as shown in a preferred embodiment in fig8 a , has a convexity 880 that comprises at least a downward curve 890 located in at least a portion of the toe region 851 . the downward curve 890 of convexity 880 begins at , or near the vicinity of , the frontmost point 850 of the upper layer 807 and gradually and continuously descends downwardly from there through at least a portion of the toe region 851 . the portion of the upper layer 807 indicated by lines extending from , and associated with , element number 880 indicates the approximate range wherein convexity 880 is typically primarily located . convexity 880 may , or may not , be entirely located within the range indicated by the lines extending from , and associated with , element number 880 . convexity 880 , as shown in a preferred embodiment in fig8 a , is relatively shallow due to its large radius , or radii , of curvature . convexity 880 may comprise a curve or curves in addition to downward curve 890 . the radius of curvature throughout convexity 880 may be completely constant , may have one or more constant portions mixed with one or more non - constant portions , or may be completely non - constant . downward curve 890 , as well as any other curve or curves that are part of convexity 880 , may , at any point on any of those curves , have a slope somewhere between negative infinity and positive infinity and can include a slope that is zero , gradual , moderate , steep , vertical or somewhere between any of those amounts . although the downward curve 890 of convexity 880 is shown in fig8 a as beginning near the frontmost point 850 , downward curve 890 of convexity 880 may instead begin at some other location on the upper layer 807 . although convexity 880 is shown in fig8 a as ending at a location in the toe region 851 , convexity 880 may instead end at some other location on the upper layer 807 . only a portion of convexity 880 may be located in the toe region 851 . alternatively , all or substantially all of convexity 880 may be located in the toe region 851 . convexity 880 , or a portion thereof , may occupy all of the toe region 851 . alternatively , convexity 880 , or a portion thereof , may occupy a substantial portion of the toe region 851 . convexity 880 has a first wavelength and a first amplitude . the bottom surface 815 of the upper layer 807 , as shown in fig8 a , has a concavity 881 that comprises at least an upward curve 891 located in at least a portion of the toe region 851 . in this preferred embodiment , concavity 881 further comprises at least a downward curve 892 . upward curve 891 may or may not be contiguous with downward curve 892 . upward curve 891 ascends upwardly in at least a portion of the toe region 851 . downward curve 892 descends downwardly in at least a portion of the toe region 851 . the portion of the upper layer 807 indicated by lines extending from , and associated with , element number 881 indicates the approximate range wherein concavity 881 is typically primarily located . concavity 881 may , or may not , be entirely located within the range indicated by the lines extending from , and associated with , element number 881 . concavity 881 has a relatively shallow depression due to its relatively long radius , or radii , of curvature . concavity 881 may comprise a curve or curves in addition to upward curve 891 and downward curve 892 . the radius of curvature throughout concavity 881 may be completely constant , may have one or more constant portions mixed with one or more non - constant portions , or may be completely non - constant . upward curve 891 , downward curve 892 , as well as any other curve or curves that are part of concavity 881 , may , at any point on any of those curves , have a slope somewhere between negative infinity and positive infinity and can include a slope that is zero , gradual , moderate , steep , vertical or somewhere between any of those amounts . although the upward curve 891 of concavity 881 is shown in fig8 a as beginning at a location near where the toe region 851 and the middle region 852 transition into one another , the upward curve 891 of concavity 881 could instead begin at some other location on the upper layer 807 . although the upward curve 891 of concavity 881 is shown in fig8 a as ending at a location near the transition between the toe region 851 and the middle region 852 , upward curve 891 may instead end at some other location on the upper layer 807 . although the downward curve 892 of concavity 881 is shown in fig8 a as beginning near the transition between the toe region 851 and the middle region 852 and ending in the vicinity of the middle region 852 , downward curve 892 of concavity 881 may instead begin at some other location on the upper layer 807 and end at some other location on the upper layer 807 . convexity 880 may or may not be contiguous with concavity 881 . only a portion of concavity 881 may be located in the in toe region 851 . alternatively , all or substantially all of concavity 881 may be located in the toe region 851 . concavity 881 has a second wavelength that is typically different from the first wavelength of convexity 880 . concavity 881 has a second amplitude that is typically different from the first amplitude of convexity 880 . the bottom surface 815 of the upper layer 807 , as shown in fig8 a , has a convexity 882 that comprises at least a downward curve 893 located in at least a portion of the middle region 852 . in this preferred embodiment , convexity 882 further comprises at least an upward curve 894 . downward curve 893 may or may not be contiguous with upward curve 894 . downward curve 893 descends downwardly in at least a portion of the middle region 852 . upward curve 894 ascends upwardly in at least a portion of the middle region 852 . the portion of the upper layer 807 indicated by lines extending from , and associated with , element number 882 indicates the approximate range wherein convexity 882 is typically primarily located . convexity 882 may , or may not , be entirely located within the range indicated by the lines extending from , and associated with , element number 882 . convexity 882 has a relatively moderate bulge due to its relatively moderate radius , or radii , of curvature . convexity 882 may comprise a curve or curves in addition to downward curve 893 and upward curve 894 . the radius of curvature throughout convexity 882 may be completely constant , may have one or more constant portions mixed with one or more non - constant portions , or may be completely non - constant . downward curve 893 , upward curve 894 , as well as any other curve or curves that are part of convexity 882 , may , at any point on any of those curves , have a slope somewhere between negative infinity and positive infinity and can include a slope that is zero , gradual , moderate , steep , vertical or somewhere between any of those amounts . although the downward curve 893 of convexity 882 is shown in fig8 a as beginning near the middle of the middle region 852 and ending near the middle of the middle region 852 , downward curve 893 of convexity 882 may instead begin at some other location on the upper layer 807 and end at some other location on the upper layer 807 . although the upward curve 894 of convexity 882 is shown in fig8 a as beginning near the middle of the middle region 852 and ending in the middle region at a location near the heel region 853 , upward curve 894 of convexity 882 may instead begin at some other location on the upper layer 807 and end at some other location on the upper layer 807 . convexity 882 may or may not be contiguous with concavity 881 . only a portion of convexity 882 may be located in the in middle region 852 . alternatively , all or substantially all of convexity 882 may be located in the middle region 852 . convexity 882 , or a portion thereof , may occupy all of the middle region 852 . alternatively , convexity 882 , or a portion thereof , may occupy a substantial portion of the middle region 852 . convexity 882 has a third wavelength that is typically different from both the first wavelength of convexity 880 and the second wavelength of concavity 881 . convexity 882 has a third amplitude that is typically different from both the first amplitude of convexity 880 and the second amplitude of concavity 881 . the bottom surface 815 of the upper layer 807 , as shown in fig8 a , has a concavity 883 that comprises at least an upward curve 895 located in at least a portion of the heel region 853 . in this preferred embodiment , concavity 883 further comprises at least a downward curve 896 . upward curve 895 may or may not be contiguous with downward curve 896 . upward curve 895 ascends upwardly in at least a portion of the heel region 853 . downward curve 896 descends downwardly in at least a portion of the heel region 853 . the portion of the upper layer 807 indicated by lines extending from , and associated with , element number 883 indicates the approximate range wherein concavity 883 is typically primarily located . concavity 883 may , or may not , be entirely located within the range indicated by the lines extending from , and associated with , element number 883 . concavity 883 has a relatively moderate depression due to its relatively moderate radius , or radii , of curvature . concavity 883 may comprise a curve or curves in addition to upward curve 895 and downward curve 896 . the radius of curvature throughout concavity 883 may be completely constant , may have one or more constant portions mixed with one or more non - constant portions , or may be completely non - constant . upward curve 895 , downward curve 896 , as well as any other curve or curves that are part of concavity 883 , may , at any point on any of those curves , have a slope somewhere between negative infinity and positive infinity and can include a slope that is zero , gradual , moderate , steep , vertical or somewhere between any of those amounts . although the upward curve 895 of concavity 883 is shown in fig8 a as beginning at a location in the middle region 852 , the upward curve 895 of concavity 883 could instead begin at some other location on the upper layer 807 . although the upward curve 895 of concavity 883 is shown in fig8 a as ending at a location near the middle of the heel region 853 of the upper layer 807 , upward curve 895 may instead end at some other location on the upper layer 807 . although the downward curve 896 of concavity 883 is shown in fig8 a as beginning near the middle of the heel region 853 and ending in the vicinity of the rearmost point 854 of the upper layer 807 , downward curve 896 of concavity 883 may instead begin at some other location on the upper layer 807 and end at some other location on the upper layer 807 . convexity 882 may or may not be contiguous with concavity 883 . only a portion of concavity 883 may be located in the in heel region 853 . alternatively , all or substantially all of concavity 883 may be located in the heel region 853 . concavity 883 , or a portion thereof , may occupy all of the heel region 853 . alternatively , concavity 883 , or a portion thereof , may occupy a substantial portion of the heel region 853 . concavity 883 has a fourth wavelength that is typically different from the first wavelength of convexity 880 , the second wavelength of concavity 881 , and the third wavelength of convexity 882 . concavity 883 has a fourth amplitude that is typically different from the first amplitude of convexity 880 , the second amplitude of concavity 881 , and the third amplitude of convexity 882 . as further shown in the embodiment in fig8 , the top surface 817 of the lower layer 809 of the midsole 803 is in substantially continuous contact with the bottom surface 815 of the upper layer 807 of the midsole . due to this substantially continuous contact between top surface 817 and bottom surface 815 in this embodiment , each convexity in the bottom surface 815 has a corresponding concavity in the top surface 817 and each concavity in the bottom surface 815 has a corresponding convexity in the top surface 817 . in other embodiments , such substantially continuous contact between top surface 817 and bottom surface 815 may not be present . the outsole 805 has a top surface 819 and a bottom surface 823 . the outsole 805 may curve upwardly in the heel region 873 . when the shoe is in its typical upright , unloaded state , the frontmost point 870 is relatively high above the ground . in this embodiment , from a point at or near the vicinity of the frontmost point 870 , the outsole 805 has a gradual downward curve 897 that continues through at least a portion of the toe region 861 of the outsole 805 , then continues to curve gradually downward in the middle region 872 of the outsole and then begins to curve upwardly forming an upward curve 898 in the heel region 873 of the outsole 805 . this gradual upward curve 898 typically continues until the outsole 805 approaches the vicinity of the rear tip 842 of the shoe . this upward curve 898 is typically sharper than the curve in the toe region 871 . upward curve 898 may be substantially sharper than shown in fig8 a or substantially shallower than shown in fig8 a . the bottom surface 823 of the outsole 805 typically contains grooves and / or patterns for optimal traction and wear . fig9 shows a front elevation view in cross section of the midsole 803 shown in fig8 along line 9 - 9 in the direction of the appended arrows . as shown in fig9 , the bottom surface 815 of the upper layer 807 is in substantially continuous contact with the top surface 817 of the lower layer 809 . the cross sectional shape of the bottom surface 815 and the top surface 817 at line 9 - 9 is shown in fig9 by a substantially horizontal line that extends from the lateral side of the midsole 803 to the medial side . fig1 shows a front elevation view in cross section of the midsole 803 shown in fig8 along line 10 - 10 in the direction of the appended arrows . as shown in fig1 , the bottom surface 815 of the upper layer 807 is in substantially continuous contact with the top surface 817 of the lower layer 809 . the cross sectional shape of the bottom surface 815 and the top surface 817 at line 10 - 10 is shown in fig1 by a substantially horizontal line that extends from the lateral side of the midsole 803 to the medial side . fig1 shows a front elevation view in cross section of the midsole 803 shown in fig8 along line 11 - 11 in the direction of the appended arrows . as shown in fig1 , the bottom surface 815 of the upper layer 807 is in substantially continuous contact with the top surface 817 of the lower layer 809 . the cross sectional shape of the bottom surface 815 and the top surface 817 at line 11 - 11 is shown in fig1 by a substantially horizontal line that extends from the lateral side of the midsole 803 to the medial side . as shown in fig9 - 11 , the midsole 803 is of varying thickness , with there generally being a progression in thickness as the midsole 803 moves from the toe region 851 to the heel region 853 . in normal use of the shoe , the user steps forward with the rear portion of the user &# 39 ; s heel stepping on the ground first . when this happens , the lower layer 809 of the midsole 803 in the heel region 853 that is made of less dense and more readily compressible material , compresses and deforms , causing the heel of the user &# 39 ; s foot to sink toward the ground to a greater extent than it would sink while wearing a conventional shoe . due to the concavity 883 , the lower layer 809 is relatively thick in the heel region 863 . since this relatively thick heel region 863 of the lower layer 809 is also relatively soft and highly compressible , it mimics the effect of walking on a sandy beach , thereby requiring the user to exert more energy while walking than would be required when walking while wearing conventional shoes . additionally , since the heel region 863 of the lower layer 809 is relatively thick and highly compressible , it has a degree of inherent lateral and transverse instability that is not present in conventional shoes . this inherent instability forces the user to make a balancing effort and use muscles and muscle control and coordination to maintain a normal walking gait that would not be required with conventional shoes . as the step continues , the user &# 39 ; s weight shifts to the center of the shoe and the shoe rolls forward in a smooth motion without the user having to overcome any abrupt pivot points . the lower layer 809 of the midsole 803 in the middle region 862 and then in the toe region 861 , compresses and deforms , allowing the user &# 39 ; s foot in those regions to sink toward the ground more than it would sink if the user were wearing conventional shoes . the convexities 880 , 882 in the toe region 861 and / or middle region 862 , limit compression of the lower layer 809 in those areas and thereby provide stability . the user then completes the step by pushing off with the forefoot ball region of the user &# 39 ; s foot . all of this simulates the effects and the fitness benefits of walking on a sandy beach or on a giving or uneven soft surface regardless of the actual hardness of the surface . while the foregoing detailed description sets forth exemplary embodiments of a shoe in accordance with the present invention , it is to be understood that the above description is illustrative only and not limiting of the disclosed invention . indeed , it will be appreciated that the embodiments discussed above and the virtually infinite embodiments that are not mentioned could easily be within the scope and spirit of the present invention .
the present invention provides a shoe having a multi - layer , multi - density midsole where the surfaces between midsole layers have one or more convexities and one or more concavities which collectively contribute to simulating the effect , and imparting the fitness benefits , of walking on a sandy beach or on a giving or uneven surface regardless of the actual hardness of the surface .