Realizability and inscribability for simplicial polytopes via nonlinear optimization

Abstract

We show that nonlinear optimization techniques can successfully be applied to realize and to inscribe matroid polytopes and simplicial spheres. Thus we obtain a complete classification of neighborly polytopes of dimension $4$, $6$ and $7$ with $11$ vertices, of neighborly $5$-polytopes with $10$ vertices, as well as a complete classification of simplicial $3$-spheres with $10$ vertices into polytopal and non-polytopal spheres. Surprisingly many of the realizable polytopes are also inscribable.

23 pages

References in corpus (3)

• Combinatorial 3-manifolds with 10 vertices
• Computing maximal copies of polytopes contained in a polytope
• NP-completeness of Partial Chirotope Extendibility

Cited by in corpus (5)

• Scribability problems for polytopes
• There are 174 Subdivisions of the Hexahedron into Tetrahedra
• Average Betti numbers of induced subcomplexes in triangulations of manifolds
• The lower bound for the number of facets of a k-neighborly d-polytope with d+3 vertices
• A Stable-Set Bound and Maximal Numbers of Nash Equilibria in Bimatrix Games