Kissing polytopes

TL;DR

This work studies the minimal distance between two disjoint lattice polytopes contained in a fixed hypercube, introducing the notion of lattice $(d,k)$-polytopes and linking the distance to optimization-complexity bounds. It establishes a near-tight dichotomy: a general lower bound $d(P,Q) \ge \frac{1}{(kd)^{2d}}$ (with stronger variants) and constructive upper bounds showing $d(P,Q) \le \frac{1}{(k\sqrt{d})^{\sqrt{d}}}$ for large $d$, with extensions to rational polytopes described via encoding length. The paper also explores special-case bounds, fixed-dimension phenomena, and computational methods for determining the exact kissing distance $\varepsilon(d,k)$, complemented by encoding-length based estimates for rational polytopes. Collectively, the results inform fundamental limits on how close disjoint polytopes can lie, with implications for linear optimization and facial-vertex distance notions.

Abstract

We investigate the following question: how close can two disjoint lattice polytopes contained in a fixed hypercube be? This question stems from various contexts where the minimal distance between such polytopes appears in complexity bounds of optimization algorithms. We provide nearly matching lower and upper bounds on this distance and discuss its exact computation. We also give similar bounds in the case of disjoint rational polytopes whose binary encoding length is prescribed.

Figures (3)

• Figure 1: Two $0/1$-polytopes $P$ and $Q$ and points $p$ and $q$ such that $d(P,Q)$ is equal to $d(p,q)$.
• Figure 2: The construction of Lemma \ref{['DHLOPP.sec.3.lem.1']} when $d$ is equal to $4$. The cube at the top is the facet of the hypercube $[0,1]^4$ made up of the points $x$ such that $x_4=0$ and the cube at the bottom is the opposite facet of $[0,1]^4$.
• Figure 3: Two polytopes $P$ and $Q$ whose distance is equal to $\varepsilon(2,k)$ (top) and $\varepsilon(3,k)$ (bottom) when $1\leq{k}\leq3$.

Theorems & Definitions (25)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Lemma 2.1
• Remark 2.2
• Theorem 2.3
• Lemma 2.4
• Theorem 3.1
• Theorem 3.2
• Proposition 3.3