Optimal tiling of the Euclidean space using symmetric bodies

TL;DR

This paper investigates the minimal surface area of symmetric bodies that tile Euclidean space, establishing tight bounds and exploring implications for symmetric parallel repetition in theoretical computer science.

Contribution

It introduces tight bounds on the surface area of symmetric tiling bodies in Euclidean space and connects these results to symmetric parallel repetition problems.

Findings

Symmetric tiling bodies have surface area at least  n/  g n

Existence of symmetric tiling bodies with surface area O(n/  g n

Results imply potential for strong parallel repetition in specific cases

Abstract

What is the least surface area of a symmetric body $B$ whose $\mathbb{Z}^n$ translations tile $\mathbb{R}^n$? Since any such body must have volume $1$, the isoperimetric inequality implies that its surface area must be at least $\Omega(\sqrt{n})$. Remarkably, Kindler et al.\ showed that for general bodies $B$ this is tight, i.e.\ that there is a tiling body of $\mathbb{R}^n$ whose surface area is $O(\sqrt{n})$. In theoretical computer science, the tiling problem is intimately to the study of parallel repetition theorems (which are an important component in PCPs), and more specifically in the question of whether a "strong version" of the parallel repetition theorem holds. Raz showed, using the odd cycle game, that strong parallel repetition fails in general, and subsequently these ideas were used in order to construct non-trivial tilings of $\mathbb{R}^n$. In this paper, motivated by the study of a symmetric parallel repetition, we consider the symmetric variant of the tiling problem in $\mathbb{R}^n$. We show that any symmetric body that tiles $\mathbb{R}^n$ must have surface area at least $\Omega(n/\sqrt{\log n})$, and that this bound is tight, i.e.\ that there is a symmetric tiling body of $\mathbb{R}^n$ with surface area $O(n/\sqrt{\log n})$. We also give matching bounds for the value of the symmetric parallel repetition of Raz's odd cycle game. Our result suggests that while strong parallel repetition fails in general, there may be important special cases where it still applies.