The lower bound for the number of facets of a k-neighborly d-polytope with d+3 vertices

TL;DR

This paper determines the minimal difference between the number of facets and vertices of k-neighborly d-polytopes with d+3 vertices, providing exact values for specific k and a formula for larger k.

Contribution

It establishes the exact minimal difference for k-neighborly d-polytopes with d+3 vertices, including a general formula for k ≥ 4.

Findings

(0)(P) = d+3 for the polytopes studied

_{d-1}(P) - _{0}(P) minimized for specific k values

A closed-form formula for _{d-1}(P) - _{0}(P) when k  4

Abstract

We have found the minimal difference $\Delta(k) = \min\limits_P (f_{d-1}(P) - f_{0}(P))$ between the number of facets and the number of vertices of a $k$-neighborly $d$-polytope $P$ for the case $f_{0}(P) = d+3$: $\Delta(2) = 4$, $\Delta(3) = 15$, and $\Delta(k) = 2 (k^2 - 1)$ for $k \ge 4$.