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Do Minkowski averages get progressively more convex?

Abstract : Let us define, for a compact set A ⊂ R n , the Minkowski averages of A: A(k) = a 1 + · · · + a k k : a 1 ,. .. , a k ∈ A = 1 k A + · · · + A k times. We study the monotonicity of the convergence of A(k) towards the convex hull of A, when considering the Hausdorff distance, the volume deficit and a non-convexity index of Schneider as measures of convergence. For the volume deficit, we show that monotonicity fails in general, thus disproving a conjecture of Bobkov, Madiman and Wang. For Schneider's non-convexity index, we prove that a strong form of monotonicity holds, and for the Hausdorff distance, we establish that the sequence is eventually nonincreasing.
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Matthieu Fradelizi, Mokshay Madiman, Arnaud Marsiglietti, Artem Zvavitch. Do Minkowski averages get progressively more convex?. Comptes Rendus. Mathématique, Académie des sciences (Paris), 2016, 354, pp.185 - 189. ⟨10.1016/j.crma.2015.12.005⟩. ⟨hal-01590072⟩



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