Random points on the unit sphere

Question

Suppose you have $n$ points picked uniformly at random on the surface of $\mathbb{S}^d,$ and let the volume of the convex hull of these points be $V_{n, d}.$ Clearly, $V_{n, d}$ converges to the volume of the unit ball in $\mathbb{R}^{d+1}$ as $n$ goes to infinity, but what is the distribution (or at least the expectation) of the difference? For $d=1,$ it is a simple computation that the $V_{n, 1} - \pi = O(1/n^3)\dots$

(there seem to be a number of questions and references about points in polygons, but I seem to be failing to find anything on spheres...)

Answer 1

As mentioned in the comments, this question has been answered for random pointes on the boundary of convex bodies and even better for all intrinsic volumes. Let me offer some references:

A good refernce is:

Matthias Reitzner, Random points on the boundary of smooth convex bodies, Trans. Amer. Math. Soc. 354, 2243-2278, 2002

Abstract:

The convex hull of $n$ independent random points chosen on the boundary of a convex body $K \subset \mathbb{R}^d$ according to a given density function is a random polytope. The expectation of its $i$-th intrinsic volume for $i=1, \dots, d$ is investigated. In the case that the boundary of $K$ is sufficiently smooth, asymptotic expansions for these expected intrinsic volumes as $n \to \infty$ are derived.

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By Ross M. Richardson, Van H. Vu and Lei Wu there are two papers, which are very simlilar:

Random inscribing polytopes, European Journal of Combinatorics. Volume 28, Issue 8, Pages 2057–2071, November 2007

and

An Inscribing Model for Random Polytopes, Discrete & Computational Geometry, Volume 39, Issue 1-3, pp 469-499, March 2008

With the following abstract:

For convex bodies $K$ with $\mathcal{C}^2$ boundary in $\mathbb{R}^d$ , we explore random polytopes with vertices chosen along the boundary of $K$. In particular, we determine asymptotic properties of the volume of these random polytopes. We provide results concerning the variance and higher moments of this functional, as well as an analogous central limit theorem.

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Another more recent reference is

Károly J. Böröczky, Ferenc Fodor, Daniel Hug, Intrinsic volumes of random polytopes with vertices on the boundary of a convex body, Trans. Amer. Math. Soc. 365, 785-809, 2013, arxiv link

Let $K$ be a convex body in $\mathbb{R}^d$, let $j\in\{1, ..., d-1\}$, and let $\varrho$ be a positive and continuous probability density function with respect to the $(d-1)$-dimensional Hausdorff measure on the boundary $\partial K$ of $K$. Denote by $K_n$ the convex hull of $n$ points chosen randomly and independently from $\partial K$ according to the probability distribution determined by $\varrho$. For the case when $\partial K$ is a $C^2$ submanifold of $\mathbb{R}^d$ with everywhere positive Gauss curvature, M. Reitzner proved an asymptotic formula for the expectation of the difference of the $j$th intrinsic volumes of $K$ and $K_n$, as $n\to\infty$. In this article, we extend this result to the case when the only condition on $K$ is that a ball rolls freely in $K$.

Answer 2

Average area (over $100$ trials) of $n$ points uniformly distributed on a unit-radius circle:

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One $n=6$ example:

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