I would like to know for which convex polyhedra $P$ in $\mathbb{R}^3$, is the center of the largest sphere enclosed in $P$ (a.k.a. the Chebyshev center, or the incenter) the same as the center of mass/gravity of $P$, assuming $P$ is a homogenous solid. The conditions force quite a bit of symmetry. Perhaps only the Platonic and Archimedean solids have the property.
The question can be asked in all dimensions, and I do not even know the answer for $\mathbb{R}^2$.
I feel this question must have been explored, but I have not found it my literature searches. I'd appreciate any pointers. Thanks!
Edit. Here are the examples suggested by Rahul and Eric. The left image is from Wikipedia's page on bipyramids. The right image I made myself.