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Workshop

Convex Cones Spanned by Regular Polytopes, and Probabilistic Applications

  • Hauke Seidel (Universität Münster, Münster, Germany)
E1 05 (Leibniz-Saal)

Abstract

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.

Links

Saskia Gutzschebauch

Max-Planck-Institut für Mathematik in den Naturwissenschaften Contact via Mail

Tim Seynnaeve

Max Planck Institute for Mathematics in the Sciences, Leipzig

Rodica Dinu

University of Bucharest

Giulia Codenotti

Freie Universität Berlin

Frank Röttger

Otto-von-Guericke-Universität, Magdeburg