Characterization of polytopes via tilings with similar pieces

2010/11/21 by Karim Adiprasito, Adiprasito, Karim
Mathematics · Materials Science · #Point processes and geometric inequalities #Advanced Combinatorial Mathematics #Quasicrystal Structures and Properties

paper · pdf · doi:10.48550/arxiv.1011.4651

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.

Related

• On some characterizations of convex polyhedra
• Translational tilings by a polytope, with multiplicity
• Twofold Translative Tilings with Convex Bodies
• Tiling Problem: Convex Pentagons for Edge-to-Edge Monohedral Tiling and Convex Polygons for Aperiodic Tiling
• Thrifty approximations of convex bodies by polytopes