When we consider a system of $N\gg 1$ one-electrons the occupied region in $k$-space is stated to be indistinguishable from a sphere, since the energy of a one-electron level is directly proportional to $k^2$.
I would rather expect this argument when the energy is proportional to the cube $k^3$ of its wave vector, since a volume in $k$-space goes as $k^3$.
How does $E(k) \sim k^2$ imply that the volume is indistinghuishable from a sphere?
Note. Ashcroft and Mermin say: If it were not spherical it would not be the ground state, for we could then construct a state of lower energy by moving the electrons in levels farthest away from $\mathbf k = 0$ into unoccupied levels closer to the origin.