Prodsimplicial-Neighborly Polytopes

TL;DR

This paper introduces PSN polytopes, a new class that generalizes neighborly and neighborly cubical polytopes, with constructions and minimal dimension bounds based on topological obstructions.

Contribution

The paper presents three methods for constructing PSN polytopes and establishes minimal dimension bounds using topological obstructions for these projections.

Findings

PSN polytopes generalize neighborly properties

Constructed via three different methods

Minimal dimension bounds established

Abstract

Simultaneously generalizing both neighborly and neighborly cubical polytopes, we introduce PSN polytopes: their k-skeleton is combinatorially equivalent to that of a product of r simplices. We construct PSN polytopes by three different methods, the most versatile of which is an extension of Sanyal and Ziegler's "projecting deformed products" construction to products of arbitrary simple polytopes. For general r and k, the lowest dimension we achieve is 2k+r+1. Using topological obstructions similar to those introduced by Sanyal to bound the number of vertices of Minkowski sums, we show that this dimension is minimal if we additionally require that the PSN polytope is obtained as a projection of a polytope that is combinatorially equivalent to the product of r simplices, when the dimensions of these simplices are all large compared to k.