Twofold Translative Tilings with Convex Bodies

TL;DR

This paper proves that for convex bodies, being a twofold translative tile is equivalent to being a onefold translative tile, clarifying the relationship between multiple and single tilings.

Contribution

It establishes a precise characterization of convex bodies that can form twofold translative tilings, showing they are exactly the ones that can form single translative tilings.

Findings

Twofold translative tilings coincide with translative tilings for convex bodies.

The result simplifies understanding of multiple tilings in convex geometry.

It provides a complete characterization of convex bodies with twofold translative tilings.

Abstract

Let $K$ be a convex body. It is known that, in general, if $K$ is a $k$-fold translative tile (for some positive integer $k$), then $K$ may not be a (onefold) translative tile. However, in this paper I will show that for every convex body $K$, $K$ is a twofold translative tile if and only if $K$ is a translative tile.