Inscribable fans, zonotopes, and reflection arrangements

Abstract: 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, universality theorems by Mnev/Richter-Gebert render the question for inscribable combinatorial types hopeless.

However, for a given complete fan N, we can decide in polynomial time if there is an inscribed polytope with normal fan N. 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 the classification of simplicial arrangements. The talk is based on joint work with Raman Sanyal.

computational geometrydiscrete mathematicscommutative algebracombinatorics

Audience: researchers in the topic

Copenhagen-Jerusalem Combinatorics Seminar

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