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A refined lower bound theorem for $d$-polytopes with at most $2d$ vertices

Guillermo Pineda-Villavicencio, Jie Wang, David Yost

TL;DR

This work deepens the understanding of sharp lower bounds on the number of $k$-faces for $d$-polytopes with $d+s$ vertices by refining Xue's bound to $\zeta_k(d+s,d)=\phi_k(d+s,d)+\binom{d-1}{k-1}-\binom{d+1-s}{k-s+1}$ whenever $s\ge3$, and detailing the $s=2$ case separately. The authors develop and leverage wedge and pyramid constructions, along with a careful inductive argument on dimension, to characterize minimisers and prove the lower bounds for polytopes with at least $d+3$ facets; in many instances only two minimisers realise the bound, though up to five can occur for some $(s,k)$. They identify the two fundamental minimisers—$T_A$-type pyramids and pyramid over the $(s-1)$-pentasm $P m(s-1,d+1-s)$—and show that their $f$-vectors coincide with $\zeta_k(d+s,d)$; in certain cases additional minimisers such as wedges $W P$, $\Sigma(3)$-based constructions, or $Z(d)$-type polytopes appear. The paper also provides a thorough equality analysis, listing the combinatorial configurations that achieve equality and offering a detailed taxonomy of minimising polytopes.

Abstract

In 1967, Grünbaum conjectured that the function $$ φ_k(d+s,d):=\binom{d+1}{k+1}+\binom{d}{k+1}-\binom{d+1-s}{k+1},\; \text{for $2\le s\le d$} $$ provides the minimum number of $k$-faces for a $d$-dimensional polytope (abbreviated as a $d$-polytope) with $d+s$ vertices. In 2021, Xue proved this conjecture for each $k\in[1\ldots d-2]$ and characterised the unique minimisers, each having $d+2$ facets. In this paper, we refine Xue's theorem by considering $d$-polytopes with $d+s$ vertices ($2\le s\le d$) and at least $d+3$ facets. If $s=2$, then there is precisely one minimiser for many values of $k$. For other values of $s$, the number of $k$-faces is at least $φ_k(d+s,d)+\binom{d-1}{k}-\binom{d+1-s}{k}$, which is met by precisely two polytopes in many cases, and up to five polytopes for certain values of $s$ and $k$. We also characterise the minimising polytopes.

A refined lower bound theorem for $d$-polytopes with at most $2d$ vertices

TL;DR

This work deepens the understanding of sharp lower bounds on the number of -faces for -polytopes with vertices by refining Xue's bound to whenever , and detailing the case separately. The authors develop and leverage wedge and pyramid constructions, along with a careful inductive argument on dimension, to characterize minimisers and prove the lower bounds for polytopes with at least facets; in many instances only two minimisers realise the bound, though up to five can occur for some . They identify the two fundamental minimisers—-type pyramids and pyramid over the -pentasm —and show that their -vectors coincide with ; in certain cases additional minimisers such as wedges , -based constructions, or -type polytopes appear. The paper also provides a thorough equality analysis, listing the combinatorial configurations that achieve equality and offering a detailed taxonomy of minimising polytopes.

Abstract

In 1967, Grünbaum conjectured that the function provides the minimum number of -faces for a -dimensional polytope (abbreviated as a -polytope) with vertices. In 2021, Xue proved this conjecture for each and characterised the unique minimisers, each having facets. In this paper, we refine Xue's theorem by considering -polytopes with vertices () and at least facets. If , then there is precisely one minimiser for many values of . For other values of , the number of -faces is at least , which is met by precisely two polytopes in many cases, and up to five polytopes for certain values of and . We also characterise the minimising polytopes.
Paper Structure (4 sections, 16 theorems, 26 equations, 2 figures)

This paper contains 4 sections, 16 theorems, 26 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Minimisers
  3. Proof of the main theorem
  4. Equality analysis

Key Result

Theorem 1

Let $d\geqslant 2$ and $1\leqslant s \leqslant d$. If $P$ is a $d$-polytope with $d+s$ vertices, then Furthermore, for some $k\in[1\ldots d-2]$, there is a unique polytope whose number of $k$-faces equals $\phi_k(d+s,d)$, and this minimiser has $d+2$ facets.

Figures (2)

  • Figure 1: Schlegel digrams of polytopes. (a) The tetragonal antiwedge. (b) The 4-polytope $W P$. (c) The 4-polytope $T A(4)$. (d) The 4-polytope $Z(4)$.
  • Figure 2: $F$ is a prism.

Theorems & Definitions (31)

  • Theorem 1: $d$-polytopes with at most $2d$ vertices, Xue 2021
  • Theorem 2: Refined theorem for $d$-polytopes with at most $2d$ vertices (Short version)
  • Lemma 3: McMullen and Shephard 1970
  • Lemma 4
  • Lemma 5
  • Theorem 6: PinYos22, Xue22
  • Lemma 7
  • Remark 8: Structure of $T A(d)$
  • Remark 9: Structure of $Z(d)$
  • Lemma 10
  • ...and 21 more