Conical equipartitions of mass distributions

Abstract

A conical dissection of Rd is a decomposition of the space into polyhedral cones. An example of a conical dissection is a fan associated to the faces of a convex polytope. Motivated by some recent questions and results about (simultaneous) conical partitions of measures by Kaneko and Kano, Bárány and Matoušek, and Bespamyatnikh et al. [2], [4], [19], we study related partition problems in higher dimensions. In the case of a single measure, several conical partition results associated to a nondegenerated pointed simplex (Δ, a) in Rn are obtained with the aid of the Brouwer fixed point theorem. In the other direction, it is demonstrated that general "symmetrical" equipartition results [21] may be used to yield, by appropriate specialization, fairly general "asymmetric," conical equipartitions for two or more mass distributions. Finally, the topological nature of these results is exemplified by their extension to the case of topological (projective) planes.