On Wendel’s equality for intersections of balls

Abstract

AbstractWe study the analogue of Wendel’s equality in random polytope models in which the hull of the random points is formed by intersections of congruent balls, called the spindle (or hyper-) convex hull. According to the classical identity of Wendel the probability that the origin is contained in the (linear) convex hull of n i.i.d. random points distributed according to an origin symmetric probability distribution in the d-dimensional Euclidean space $$\mathbb {R}^{d}$$ R d that assigns measure zero to hyperplanes is a constant depending only on n and d. While in the classical convex case one gets nonzero probabilities only for $$n\ge d+1$$ n ≥ d + 1 points in $$\mathbb {R}^{d}$$ R d , for the spindle convex hull this happens for all $$n\ge 2$$ n ≥ 2 . We study this question for the uniform and normally distributed random models.