Characterization of polytopes via tilings with similar pieces

TL;DR

This paper proves that a convex body is a polytope if it admits multiple tilings with tiles similar to itself, extending previous results and providing a counterexample to potential improvements.

Contribution

The paper generalizes existing theorems by showing that multiple tilings with similar tiles characterize polytopes, and it demonstrates the limits of this characterization with a counterexample.

Findings

Convex body is a polytope if many tilings contain similar tiles.

Counterexample shows the result cannot be strengthened.

Extension of Valette, Zamfirescu, and Laczkovich's results.

Abstract

Generalizing results by Valette, Zamfirescu and Laczkovich, we will prove that a convex body $K$ is a polytope if there are sufficiently many tilings which contain a tile similar to $K$. Furthermore, we give an example that this can not be improved.