TY - JOUR
T1 - Dual affine invariant points
AU - Meyer, Mathieu
AU - Schütt, Carsten
AU - Werner, Elisabeth
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
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U2 - 10.1512/iumj.2015.64.5514
DO - 10.1512/iumj.2015.64.5514
M3 - Article
AN - SCOPUS:84956664431
SN - 0022-2518
VL - 64
SP - 735
EP - 768
JO - Indiana University Mathematics Journal
JF - Indiana University Mathematics Journal
IS - 3
ER -