Flat simplices and kissing polytopes

TL;DR

The paper investigates how flat a lattice $(d,k)$-simplex can be inside the hypercube $[0,k]^d$ by studying the minimal distance $\varepsilon_i(d,k)$ between opposite faces and the kissing-distance $\varepsilon(d,k)$. It develops an algebraic framework using a $d\times(d-1)$ matrix $A$ and an integer vector $b$ to express the unbounded distance $\varepsilon_i^u(d,k)$ as $\|A(A^T A)^{-1}A^Tb-b\|$, and establishes monotonicity properties and relations to projections. The authors derive improved universal lower bounds on $\varepsilon(d,k)$ via bounds on $\det(A^T A)$, including a sharper bound when $i=0$ and $k=1$, and show that for $d=3$ the exact value $\varepsilon_0(3,k)=\frac{1}{\sqrt{3k^4-4k^3+4k^2-2k+1}}$ holds for all $k\ge 2$, with a reduction to a finite combinatorial set and a unique extremal configuration up to symmetry. These results advance the understanding of kissing polytopes and flat simplices in lattice polytopes, providing explicit formulas and a scalable computational approach.

Abstract

We consider how flat a lattice simplex contained in the hypercube $[0,k]^d$ can be. This question is related to the notion of kissing polytopes: two lattice polytopes contained in the hypercube $[0,k]^d$ are kissing when they are disjoint but their distance is as small as possible. We show that the smallest possible distance of a lattice point $P$ contained in the cube $[0,k]^3$ to a lattice triangle in the same cube that does not contain $P$ is $$ \frac{1}{\sqrt{3k^4-4k^3+4k^2-2k+1}} $$ when $k$ is at least $2$. We also improve the known lower bounds on the distance of kissing polytopes for $d$ at least $4$ and $k$ at least $2$.

Figures (1)

• Figure 1: A lattice point and a lattice triangle that achieve $\varepsilon_0(3,k)$ for $k$ equal to $1$, $2$, and at least $3$ (from left to right).

Theorems & Definitions (23)

• Theorem 1.1
• Theorem 1.2
• Theorem 2.1
• Remark 2.2
• Corollary 2.3
• Remark 2.4
• Theorem 2.5
• Remark 2.6
• Lemma 2.7
• Remark 2.8