Journal of Convex Analysis 32 (2025), No. 1, 061--070
Copyright Heldermann Verlag 2025
B-Convexity, Convexification of Minkowski Averages in a Banach Space, and SLLN for Random Sets
Zvi Artstein
Department of Mathematics, The Weizmann Institute of Science, Rehovot, Israel
zvi.artstein@weizmann.ac.il
Vladimir Kadets
School of Mathematical Sciences, Holon Institute of Technology, Holon, Israel
vova1kadets@yahoo.com
For an infinite-dimensional Banach space X, we demonstrate the equivalence of the following two properties. One, the space is B-convex, that is, it possesses a nontrivial type. Two, X possesses the convexification property, that is, the Hausdorff distance between the Minkowski average of k subsets of the unit ball, and the convex hull of the average, converges to 0 as k tends to infinity. A rate for the convergence is provided. The result is used to establish a general Strong Law of Large Numbers for random bounded subsets of the Banach space.
Keywords: Banach space, Minkowski averages, convexification, random sets.
MSC: 46B20, 28B20.
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