Convex Cones Spanned by Regular Polytopes, and Probabilistic Applications

Let $P$ be an $n$-dimensional regular crosspolytope, simplex, or cube centred at the origin of ${\mathbb{R}}^n$. We consider convex cones of the form $$C=\{\lambda x+\lambda e_{n+1}:\lambda \ge 0,x\in P\}\subset {\mathbb{R}}^{n+1},$$ where $e_1,\dots ,e_{n+1}$ is the standard basis of ${\mathbb{R}}^{n+1}$.

We shall derive explicit probabilistic expressions for the inner and outer solid angles of these cones. As a corollary, we shall derive a formula for the inner and outer solid angles of a regular crosspolytope. Furthermore, we shall compute the probability that a random linear subspace intersects a fixed face of a regular crosspolytope, cube or simplex.

As another application, we shall explain how these cones are an important tool in determining the absorption probability of the symmetric Gaussian polytope ${\mathcal{P}}_{n,d}$, that is the probability that a deterministic point $x\in {\mathbb{R}}^d$ is contained in the convex hull of $n$ independent standard normally distributed points $X_1,\dots ,X_n$ in ${\mathbb{R}}^d$ together with their negatives $-X_1,\dots ,-X_n$.

Only few stochastic knowledge is necessary to follow this talk, since the methods presented are relatively basic.