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arXiv · 1310.0128

Dual Affine invariant points

Mathieu Meyer·Carsten Schuett·Elisabeth M. Werner
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Abstract

An affine invariant point on the class of convex bodies in R^n, endowed with the Hausdorff metric, is a continuous map p which is invariant under one-to-one affine transformations A on R^n, that is, p(A(K))=A(p(K)). We define here the new notion of dual affine point q of an affine invariant point p by the formula q(K^{p(K)})=p(K) for every convex body K, where K^{p(K)} denotes the polar of K with respect to p(K). We investigate which affine invariant points do have a dual point, whether this dual point is unique and has itself a dual point. We define a product on the set of affine invariant points, in relation with duality. Finally, examples are given which exhibit the rich structure of the set of affine invariant points.

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Mathieu Meyer, Carsten Schuett, Elisabeth M. Werner. 2013-10-01. Dual Affine invariant points. https://arxiv.org/abs/1310.0128

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