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 $$C=\{ \lambda x+\lambda e_{n+1} : \lambda \geq 0, x\in P \} \subset \mathbb{R}^{n+1}, $$ where $e_1,\ldots,e_{n+1}$ 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 $X_1,\ldots,X_n$ in $\mathbb{R}^d$ together with their negatives $-X_1,\ldots,-X_n$.

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