  We construct a new volume preserving map from the unit ball $\mathbb B^3$ to
the regular 3D octahedron, both centered at the origin, and its inverse. This
map will help us to construct refinable grids of the 3D ball, consisting in
diameter bounded cells having the same volume. On this 3D uniform grid, we
construct a multiresolution analysis and orthonormal wavelet bases of
$L^2(\mathbb B^3)$, consisting in piecewise constant functions with small local
support.
