Abstract

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.

Author information

Contact details are reproduced from the original publication and may be historical.

Vladimir Kadets

School of Mathematical Sciences, Holon Institute of Technology, Holon, Israel

vova1kadets@yahoo.com

Keywords

  • Banach space
  • Minkowski averages
  • convexification
  • random sets

Mathematics Subject Classification

46B20, 28B20

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Suggested citation

Z. Artstein, V. Kadets. “B-Convexity, Convexification of Minkowski Averages in a Banach Space, and SLLN for Random Sets.” Journal of Convex Analysis 32 (2025), No. 1, 61–70. https://doi.org/10.68381/jca32005

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Published by Heldermann Verlag, 2025. Rights now held by Banach Press LLC.