Mathematika

Research Article

On the Steinhaus tiling problem

Mihail N. Kolountzakisa1 and Thomas Wolffa2

a1 Department of Mathematics, University of Crete, 714 09 Iraklion, Crete, Greece. e-mail: kolount@math.uch.gr

a2 Department of Mathematics, 253-37 Caltech, Pasadena, CA 91125, USA. e-mail: wolff@cco.caltech.edu

Abstract

Several results are proved related to a question of Steinhaus: is there a set E2 such that the image of E under each rigid motion of IR2 contains exactly one lattice point? Assuming measurability, the analogous question in higher dimensions is answered in the negative, and on the known partial results in the two dimensional case are improved on. Also considered is a related problem involving finite sets of rotations.

(Received November 03 1997)

Key Words:

  • 52A37: CONVEX AND DISCRETE GEOMETRY; General convexity; Other problems of combinatorial convexity