Dual Affine invariant points
Functional Analysis
2013-10-02 v1
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.
Cite
@article{arxiv.1310.0128,
title = {Dual Affine invariant points},
author = {Mathieu Meyer and Carsten Schuett and Elisabeth M. Werner},
journal= {arXiv preprint arXiv:1310.0128},
year = {2013}
}