IUMJ

Title: Dual Affine invariant points

Authors: Elisabeth Werner, Mathieu Meyer and Carsten Schuett

Issue: Volume 64 (2015), Issue 3, 735-768

Abstract:

An affine invariant point on the class of convex bodies $\mathcal{K}_n$ in $\mathbb{R}^n$, endowed with the Hausdorff metric, is a continuous map from $\mathcal{K}_n$ to $\mathbb{R}^n$ that is invariant under one-to-one affine transformations $A$ on $\mathbb{R}^n$, 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 $K\in\mathcal{K}_n$, 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 also 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.