A Ham Sandwich Analogue for Quaternionic Measures and Finite Subgroups of S^3

Abstract

A "ham sandwich" theorem is established for n quaternionic Borel measures on quaternionic space H^n. For each finite subgroup G of S^3, it is shown that there is a quaternionic hyperplane H and a corresponding tiling of H^n into |G| fundamental regions which are rotationally symmetric about H with respect to G, and satisfy the condition that for each of the n measures, the "G average" of the measures of these regions is zero. If each quaternionic measure is a 4-tuple of finite Borel measures on R^{4n}, the original ham sandwich theorem on R^{4n} is recovered when G = Z_2. The theorem applies to [n/4] finite Borel measures on R^n, and when G is the quaternion group Q_8 this gives a decomposition of R^n into 2 rings of 4 cubical "wedges" each, such that the measure any two opposite wedges is equal for each finite measure.

arXiv commentThis paper has been withdrawn by the author. Now part of larger paper, Equivariant Equipartitions: Ham Sandwich Theorems for Finite Subgroups of Spheres