Matching centroids by a projective transformation

Abstract

Given two subsets of mathbb {R}^d, when does there exist a projective transformation that maps them to two sets with a common centroid? When is this transformation unique modulo affine transformations? We study these questions for 0- and d-dimensional sets, obtaining several existence and uniqueness results as well as examples of non-existence or non-uniqueness. If both sets have dimension 0, then the problem is related to the analytic center of a polytope and to polarity with respect to an algebraic set. If one set is a single point, and the other is a convex body, then it is equivalent by polar duality to the existence and uniqueness of the Santaló point. For a finite point set versus a ball, it generalizes the Möbius centering of edge-circumscribed convex polytopes and is related to the conformal barycenter of Douady-Earle. If both sets are d-dimensional, then we are led to define the Santaló point of a pair of convex bodies. We prove that the Santaló point of a pair exists and is unique, if one of the bodies is contained within the other and has Hilbert diameter less than a dimension-depending constant. The bound is sharp and is obtained by a box inside a cross-polytope.