Suppose we take the unit ball in $\mathbb{R}^n$, centered at the origin, and we start sampling points on its surface uniformly and independently at random. As an asymptotic function of $n$, how many points will we need to sample before the origin is enclosed by the convex hull of the sampled points, with probability $\ge .5$?
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4This may be related: https://math.stackexchange.com/questions/1400/probability-that-the-convex-hull-of-random-points-contains-spheres-center – Intelligenti pauca Jul 26 '17 at 15:04