Realizability and inscribability for simplicial polytopes via nonlinear optimization

 Author: Moritz Firsching Journal: Mathematical Programming, 1661--2273 -- 295, 2017. Full text: arXiv • DOI • journal

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.

@Article{ FirschingRealizability2017,
author = "Moritz Firsching",
title = "Realizability and inscribability for simplicial polytopes via nonlinear optimization",
year = "2017",
journal = "Mathematical Programming",
volume = "166",
number = "1--2",
pages = "273 -- 295",
doi = "10.1007/s10107-017-1120-0"
}