Realizability and inscribability for simplicial polytopes via nonlinear optimization

• 成果类型: Article
• 署名作者: Firsching, Moritz
• 署名单位: Free University of Berlin
• 刊物名称: MATHEMATICAL PROGRAMMING
• ISSN/ISSBN: 0025-5610
• DOI: 10.1007/s10107-017-1120-0
• 发表日期: 2017
• 页码: 273-295
• 关键词: neighborly polytopes; complete enumeration; convex; number; 3-spheres; polyhedra; 4-polytopes; bounds

摘要

We show that nonlinear optimization techniques can successfully be applied to realize and to inscribe matroid polytopes and simplicial spheres. In order to show non-realizability of simplicial spheres, we extend the method of finding biquadratic final polynomials for matroid polytopes to partial matroid polytopes. Combining these two methods 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.