Angles of orthocentric simplices

TL;DR

This paper provides explicit formulas for angles of orthocentric simplices, classifies them based on the orthocenter's position, and applies these results to compute expected face counts and volumes of certain Gaussian-related random polytopes.

Contribution

It introduces orthocentric cones, derives angle formulas for all orthocentric simplices, and applies these to probabilistic geometric problems.

Findings

Explicit angle formulas for orthocentric simplices in different cases.

Classification of orthocentric simplices as acute, rectangular, or obtuse.

Calculation of expected face numbers and volumes of Gaussian-based random polytopes.

Abstract

A $d$-dimensional simplex in Euclidean space is called orthocentric if all of its altitudes intersect at a single point, referred to as the orthocenter. We explicitly compute the internal and external angles at all faces of an orthocentric simplex. To this end, we introduce a parametric family of polyhedral cones, called orthocentric cones, and derive formulas for their angles and, more generally, for their conic intrinsic volumes. We characterize the tangent and normal cones of orthocentric simplices in terms of orthocentric cones with explicit parameters. Depending on whether the orthocenter lies inside the simplex, on its boundary, or outside, the simplex is classified as acute, rectangular, or obtuse, respectively. The solid angle formulas differ in these three cases. As a probabilistic application of the angle formulas, we explicitly compute the expected number of $k$-dimensional faces and the expected volume of the random polytope $[g_1/\tau_1, \ldots, g_n/\tau_n]$, where $g_1, \ldots, g_n$ are independent standard Gaussian vectors in $\mathbb{R}^d$, and $\tau_1, \ldots, \tau_n > 0$ are constants.