Having a cube, with a point at its center. What are the points that are equidistant from the center point to the cubes vertices? Having a cube, with a point at its center. 

What shape do the points wich are equidistant between the center and the cubes vertices make?

The source of why I had this question is the following photo



What shape is resultant from this composition of equidistant points? 

Thank you very much.
The answer to the question from screenshot is a truncated octahedron. If you consider center of the cube and a vertex, the locus of equidistant points is a perpendicular bisector plane. 8 of those planes create an octahedron. However, if you consider centers of other cubes, locus of equidistant points with them will give the initial cube itself. So the answer is the intersection of cube and octahedron, which is a truncated octahedron.
Within the pic of OP there is the quest for the volume of $D(0)$ as well. $D(0)$ was already being mentioned to be the truncated octahedron. Thus the searched for volume is $$V=8\sqrt 2\ s^3$$

where $s$ is the edge length of that polyhedron. That length in turn can be seen from that very pic to be (in units of the lattice constant $a$) $$s=\frac 12 \sqrt 2\ a$$

Thus the searched for total value would become $$V=4\ a^3$$