Angles of orthocentric simplices
Zakhar Kabluchko, Philipp Schange
TL;DR
This work provides explicit angle formulas for orthocentric simplices in $\mathbb{R}^d$ by introducing the orthocentric-cone framework and defining the function $\mathbf{g}_d$ as a Gaussian orthant probability; it gives complete classifications of acute, rectangular, and obtuse cases and expresses face-angles via $\mathbf{g}_d$ with detailed integral representations. It further develops duality and tangent/normal-cone structure within the orthocentric family, enabling analytic tractability of conic-intrinsic volumes and enabling Gaussian-projection results for random polytopes through Tsirelson-type theorems. The paper also provides explicit formulas for $\mathbf{g}_d$ in the all-positive parameter regime and in the regime with a negative parameter, along with limiting results for rectangular simplices; these developments yield exact expressions for the expected numbers of $k$-faces and volumes of Gaussian polytopes of the form $[g_1/\tau_1,\dots,g_n/\tau_n]$. Overall, the results bridge high-dimensional simplex geometry with probabilistic analysis of Gaussian polytopes, offering precise, implementable tools for computational convex geometry and stochastic geometry.
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 $[g_1/τ_1, \ldots, g_n/τ_n]$, where $g_1, \ldots, g_n$ are independent standard Gaussian vectors in $\mathbb{R}^d$, and $τ_1, \ldots, τ_n > 0$ are constants.