Steven Hoehner
Longwood University
Title
Maximum volume polyhedra inscribed in a sphere
Abstract
A classical problem in convex and discrete geometry asks: Among all convex polyhedra with a prescribed number of vertices inscribed in a sphere, which one has the largest volume? Despite its simple formulation, this problem becomes increasingly difficult as the number of vertices grows. In this talk, I will discuss recent solutions for nine and ten vertices, as well as progress on the eleven vertices case, where we determine the combinatorial structure of the maximizer. The proofs combine geometric inequalities with combinatorial arguments to eliminate competing configurations. If time permits, I will also discuss related results for polyhedra with a prescribed number of facets or edges.
