Flat simplices and kissing polytopes

TL;DR

This paper investigates the minimal distances between lattice simplices and kissing polytopes within hypercubes, providing exact formulas in three dimensions and improved bounds in higher dimensions.

Contribution

It derives an explicit formula for the minimal distance in 3D and enhances lower bounds for higher dimensions and larger hypercube sizes.

Findings

Exact minimal distance formula in 3D for k ≥ 2

Improved lower bounds for kissing polytopes in higher dimensions

Insights into the geometry of lattice simplices within hypercubes

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 $2$.