Do Minkowski averages get progressively more convex?

详细信息    查看全文

• 作者：Matthieu Fradelizi^a ; ¹ ; ^{matthieu.fradelizi@u-pem.fr" class="auth_mail" title="E-mail the corresponding author} ; Mokshay Madiman^b ; ² ; ^{madiman@udel.edu" class="auth_mail" title="E-mail the corresponding author} ; Arnaud Marsiglietti^c ; ³ ; ^{arnaud.marsiglietti@ima.umn.edu" class="auth_mail" title="E-mail the corresponding author} ; Artem Zvavitch^d ; ⁴ ; ^{zvavitch@math.kent.edu" class="auth_mail" title="E-mail the corresponding author}
• 刊名：Comptes Rendus Mathematique
• 出版年：2016
• 出版时间：February 2016
• 年：2016
• 卷：354
• 期：2
• 页码：185-189
• 全文大小：249 K

文摘

Let us define, for a compact set A⊂Rⁿ$A\subset {\mathbf{R}}^n$, the Minkowski averages of A: We study the monotonicity of the convergence of A(k)$A(k)$ 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.