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. * Supported by the DFG within SFB/TRR 109 "Discretization in Geometry and Dynamics"

A Proof of the Strict Monotone 5-Step Conjecture

“…If an orientation of a 5-polytope were a counterexample to the strictly monotone 5-step conjecture, then [8] shows that we could assume that the source and sink do not share any facets. In particular, we could assume that the source of the orientation would be on facets 1, 2, 3, 4, 5 and that its sink would be on facets 6,7,8,9,10. Such a counterexample would lead to a function χ satisfying 10 constraints each for these two vertices.…”

“…The constraints χ((1, 2, 3, 4, 5), 11) = χ((1, 2, 3, 4, 5) k,12 , 11) for k = 1, 2, 3, 4, 5, together imply that [1,2,3,4,5] is a vertex of the oriented matroid program, where (1, 2, 3, 4, 5) k,12 is the ordered set (1, 2, 3, 4, 5) with entry k replaced by 12. Similarly, the constraints χ((6, 7, 8, 9, 10), 11) = χ((6, 7, 8, 9, 10) k,12 , 11) for k = 6, 7, 8, 9, 10 imply that [6,7,8,9,10] is a vertex.…”

“…The constraints χ((1, 2, 3, 4, 5), 12) = χ((1, 2, 3, 4, 5) k,11 , 12), for k = 1, 2, 3, 4, 5, ensure that all of the edges containing vertex [1,2,3,4,5] are oriented away from that vertex, while the constraints χ((6, 7, 8, 9, 10), 12) = −χ((6, 7, 8, 9, 10) k,11 , 12), orient the edges containing [6,7,8,9,10] toward [6,7,8,9,10]. We also have to introduce constraints to avoid each possible monotone path v 0 , v 1 , .…”

A Flag Vector of a 3‐sphere That Is Not the Flag Vector of a 4‐polytope

“…The only exception to this are the triangles on the boundary, which are the facets of the convex hull of the diagram. Use the Grassmann–Plücker relations to determine further entries of the partial chirotope. The resulting partial chirotopes for the different facets as bases give the signs of 199 to 328 elements out of , which is the size of an oriented matroid for a diagram of . Using an approach recently introduced by Firsching [15], we used SCIP [1] to find coordinates for the diagrams with bases , while in the case of the diagram with base we used backtracking to find for every oriented matroid a partial chirotope of size at least 435 (i.e. of roughly 87.5%).…”

“…The resulting partial chirotopes for the different facets as bases give the signs of 199 to 328 elements out of 12 4 = 495, which is the size of an oriented matroid for a diagram of W 40 12 . Using an approach recently introduced by Firsching [15], we used SCIP [1] to find coordinates for the diagrams with bases F 1 , . .…”