An integer parallelotope with small surface area

TL;DR

This paper constructs a convex body in high-dimensional space with minimal surface area that can tile the entire space via integer lattice translations, revealing new geometric properties of such tilings.

Contribution

It demonstrates the existence of a convex body with small surface area capable of tiling space through integer lattice translations, a novel result in geometric tiling theory.

Findings

Convex body with surface area at most n^{1/2+o(1)} exists in any dimension.

Such bodies can tile space via integer lattice translations.

Provides new insights into the relationship between surface area and tiling properties.

Abstract

We prove that for any $n\in \mathbb{N}$ there is a convex body $K\subseteq \mathbb{R}^n$ whose surface area is at most $n^{\frac12+o(1)}$, yet the translates of $K$ by the integer lattice $\mathbb{Z}^n$ tile $\mathbb{R}^n$.