Hamilton Connectivity of Convex Polytopes with Applications to Their Detour Index

Asymmetry of convex polytopes and vertex index of symmetric convex

In [GL] it was shown that a polytope with few vertices is far from being symmetric in the Banach-Mazur distance. More precisely, it was shown that Banach-Mazur distance between such a polytope and any symmetric convex body is large. In this note we introduce a new, averaging-type parameter to measure the asymmetry of polytopes. It turns out that, surprisingly, this new parameter is still very l...

Applications of Convex and Algebraic Geometry to Graphs and Polytopes

On the Graph Connectivity of Skeleta of Convex Polytopes

Given a d-dimensional convex polytope P and nonnegative integer k not exceeding d − 1, let Gk(P ) denote the simple graph on the node set of k-dimensional faces of P in which two such faces are adjacent if there exists a (k + 1)-dimensional face of P which contains them both. The graph Gk(P ) is isomorphic to the dual graph of the (d− k)-dimensional skeleton of the normal fan of P . For fixed v...

Fibonacci Polytopes and Their Applications

A Fibonacci d-polytope of order k is defined as the convex hull of {0, 1}-vectors with d entries and no consecutive k ones, where k ≤ d. We show that these vertices can be partitioned into k subsets such that the convex hull of the subsets give the equivalent of Fibonacci (d− i)polytopes, for i = 1, . . . , k, which yields a “Fibonacci like” recursive formula to enumerate the vertices. Surprisi...

Convex Polytopes and the Index of Wiener–hopf Operators

We study the C∗-algebra of Wiener–Hopf operators AΩ on a cone Ω with polyhedral base P. As is known, a sequence of symbol maps may be defined, and their kernels give a filtration by ideals of AΩ, with liminary subquotients. One may define K-group valued “index maps” between the subquotients. These form the E1 term of the Atiyah–Hirzebruch type spectral sequence induced by the filtration. We sho...