Fitting centroids by a projective transformation

TL;DR

This paper investigates conditions for the existence and uniqueness of projective transformations aligning two sets in R^d to share a centroid, linking geometric concepts like the Santaló point and conformal barycenter.

Contribution

It provides new existence and uniqueness results for projective transformations aligning sets of various dimensions, extending classical geometric concepts.

Findings

Existence and uniqueness results for 0- and d-dimensional sets

Connection to Santaló point and conformal barycenter

Sharp bounds for the Santaló point of convex body pairs

Abstract

Given two subsets of 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 Santal\'o point. For a finite point set versus a ball, it generalizes the M\"obius 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 Santal\'o point of a pair of convex bodies. We prove that the Santal\'o 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.