TY - JOUR

T1 - Dual affine invariant points

AU - Meyer, Mathieu

AU - Schütt, Carsten

AU - Werner, Elisabeth

N1 - Copyright:
Copyright 2016 Elsevier B.V., All rights reserved.

PY - 2015

Y1 - 2015

N2 - An affine invariant point on the class of convex bodies Kn in Rn, endowed with the Hausdorff metric, is a continuous map from Kn to Rn that is invariant under one-to-one affine transformations A on Rn, 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(Kp(K)) = p(K) for every K Kn, where Kp(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.

AB - An affine invariant point on the class of convex bodies Kn in Rn, endowed with the Hausdorff metric, is a continuous map from Kn to Rn that is invariant under one-to-one affine transformations A on Rn, 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(Kp(K)) = p(K) for every K Kn, where Kp(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.

KW - Affine invariant point

KW - Dual affine invariant point

UR - http://www.scopus.com/inward/record.url?scp=84956664431&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84956664431&partnerID=8YFLogxK

U2 - 10.1512/iumj.2015.64.5514

DO - 10.1512/iumj.2015.64.5514

M3 - Article

AN - SCOPUS:84956664431

VL - 64

SP - 735

EP - 768

JO - Indiana University Mathematics Journal

JF - Indiana University Mathematics Journal

SN - 0022-2518

IS - 3

ER -