Which high-dimensional lattices (particularly $Z_n^*,D_n,D_n^*,A_n,A_n^*$), exhibit the following property shown in the attached diagram? Two 2D lattices are shown, with the lattice points in red, the Voronoi cells in black, and the origin cell in blue. The cells closest to the origin cell are shown in green. Shown in purple is a point that belongs to the origin cell, and is displaced a small distance to a different cell.

In the hexagonal lattice, we see that such a small displacement will only transport the purple dot to one of the closest neighboring cells (green). But in the cubic lattice, a small displacement can move the purple dot to a Voronoi cell which is not one of the closest neighbors. Which other high-dimensional lattices share this property with the cubic lattice? I have consulted Conway and Sloan's book, but this question did not appear to be addressed. If anyone knows the answer or can point me to helpful references, I'd really appreciate it.