On the minimum number of facets of a 2-neighborly polytope

TL;DR

This paper investigates the minimal number of facets in 2-neighborly polytopes, providing new bounds and exact values for various dimensions and vertex counts, advancing understanding of polytope combinatorics.

Contribution

It establishes new lower bounds and exact values for the minimal facets of 2-neighborly polytopes across different dimensions, using combinatorial and geometric methods.

Findings

Proves or 5-dimensional polytopes, ounds grow as y  nd for 6-dimensional, the minimal facets are at least the number of vertices.

The paper uses the g-theorem to derive formulas for the minimal facets of simplicial 2-neighborly polytopes, linking combinatorial invariants to geometric properties.

Abstract

Let $\mu_{\text{2n}}(d,v)$ (respectively, $\mu^{\text{s}}_{\text{2n}}(d,v)$) be the minimal number of facets of a (simplicial) 2-neighborly $d$-polytope with $v$ vertices, $v > d \ge 4$. It is known that $\mu_{\text{2n}}(4,v) = v (v-3)/2$, $\mu_{\text{2n}}(d, d+2) = d+5$, $\mu_{\text{2n}}(d,d+3) = d+7$ for $d \ge 5$, and $\mu_{\text{2n}}(d,d+4) \in [d+5, d+8]$ for $d \ge 6$. We show that $\mu_{\text{2n}}(5, v) = \Omega(v^{4/3})$, $\mu_{\text{2n}}(6, v) \ge v$, and the equality $\mu_{\text{2n}}(6, v) = v$ holds only for a simplex and for a dual 2-neighborly 6-polytope (if it exists) with $v \ge 27$. By using $g$-theorem, we get $\mu^{\text{s}}_{\text{2n}}(d, v) = \Delta (\Delta(d-3) + 3d - 5)/2 + d + 1$, where $\Delta = v - d - 1$. Also we show that $\mu_{\text{2n}}(d, v) \ge d+7$ for $v \ge d+4$.