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**1 - 3**of**3**### Dual Affine invariant points

, 2013

"... An affine invariant point on the class of convex bodies Kn in R n, endowed with the Hausdorff metric, is a continuous map from Kn to R n 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 a ..."

Abstract
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An affine invariant point on the class of convex bodies Kn in R n, endowed with the Hausdorff metric, is a continuous map from Kn to R n 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 K ∈ Kn, 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.