Dual affine invariant points

Mathieu Meyer, Carsten Schütt, Elisabeth Werner

Research output: Contribution to journalArticlepeer-review

4 Scopus citations


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.

Original languageEnglish (US)
Pages (from-to)735-768
Number of pages34
JournalIndiana University Mathematics Journal
Issue number3
StatePublished - 2015


  • Affine invariant point
  • Dual affine invariant point


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