Maximum volume polytopes inscribed in the unit sphere

Abstract

In this paper we investigate the problem of finding the maximum volume polytopes, inscribed in the unit sphere of the d-dimensional Euclidean space, with a given number of vertices. We solve this problem for polytopes with d + 2 vertices in every dimension, and for polytopes with d + 3 vertices in odd dimensions. For polytopes with d + 3 vertices in even dimensions we give a partial solution.2010 Mathematics Subject Classification. 52B60, 52A40, 52A38.

Volume of Convex Hull of Two Bodies and Related Problems

“…The first non-trivial case is when the number of points is equal to n = d + 2. It has been proved: 20]). Let P ∈ P d (d + 2) have maximal volume over P d (d + 2).…”

“…We now extract the method of Berman and Hanes to higher dimensions and using a combinatorial concept, the idea of Gale's transform solve some cases of few vertices. In this subsection we collect the results of the paper [20]. The first step is the generalization of Lemma 2.1 for arbitrary dimensions.…”

“…[20]). Let P ∈ C d (d + 3) be a cyclic polytope, where d = 4 or d = 6, and let V (P ) = {p i : i = 1, 2, .…”

A characterization of affinely regular polygons

“…We solve a more general problem: we characterize the d-dimensional polytopes with n vertices whose geometric automorphism group contains D n as a subgroup. This result can be regarded as a first step towards solving Problem 1 [11], and we note that D n is the combinatorial automorphism group of a cyclic polytope in an even dimensional space.…”

On the icosahedron inequality of László Fejes-Tóth

“…Nevertheless, according to our knowledge, this question, which is listed in both research problem books [2] and [3], is still open for polyhedra with n > 8 vertices apart from the fortunate case of n = 12 when the solution is the regular icosahedron. In [9] the authors investigated this problem for polytopes in arbitrary dimensions. By generalizing the methods of [1], presented a necessary condition for the optimality of a polytope.…”