Inscribable fans, zonotopes, and reflection arrangements
Steiner posed the question if any 3-dimensional polytope had a realization with vertices on a sphere. Steinitz constructed the first
counter example and Rivin gave a complete resolution. In dimensions 4 and up, Mnev's universality theorem renders the question
for inscribable combinatorial types hopeless.
However, for a given complete fan F, we can decide in polynomial time if there is an inscribed polytope with normal fan F. Linear
hyperplane arrangements can be realized as normal fans via zonotopes. It turns out that inscribed zonotopes are rare and in this talk I
will focus on the question of classifying the corresponding arrangements. This relates to localizatons and restrictions of
reflection arrangements and Grünbaum's quest for simplicial arrangements.
This is based on joint work with Raman Sanyal.