B-Convexity, Convexification of Minkowski Averages in a Banach Space, and SLLN for Random Sets

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.