<div class="csl-bib-body">
<div class="csl-entry">Izmestiev, I. (2023). Matching centroids by a projective transformation. <i>Geometriae Dedicata</i>, <i>217</i>(3), Article 53. https://doi.org/10.1007/s10711-023-00789-9</div>
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dc.identifier.issn
0046-5755
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dc.identifier.uri
http://hdl.handle.net/20.500.12708/191015
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dc.description.abstract
Given two subsets of ℝd, 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.