Convex side of a spherical mirror 
A convex set has the property that if you take any two points in the
  set and draw the line segment connecting those two points, that line
  segment lies entirely in the set.


My textbook says that the figure on top represents a convex mirror and that the convex side of the spherical mirror faces the incident light. However, if I take two points $A$ and $P$ on this side, the line segment $AP$ does not lie entirely on this side. How, then, can the side of this mirror that faces the incident light be called the convex side?

I agree that it is confusing that the boundary of a convex set, as seen from any point in that set, is actually concave, in the sense of curving in or hollowed inward. It might have been better if the set was called the "concave set". 
In any case, if the boundary, as seen from any point within the set, is concave, it follows that the boundary, as seen from any point outside the set, must be convex.
