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Philipp Schange

Publications and source records attributed to Philipp Schange.

4 recordsLinked to original sources
arXiv2026

Angles and volumes of regular polytopes in geometries of constant curvature

We derive closed-form expressions for the internal and external angles of $d$-dimensional cubes, regular simplices and regular crosspolytopes in geometries of constant sectional curvature $κ\in \mathbb R$. More generally, we determine internal and external angles at arbitrary faces of rectangular boxes, acute orthocentric simplices, rectangular orthocentric simplices and asymmetric crosspolytopes in arbitrary dimension $d$. We also characterize Riemannian tangent and normal cones of these polytopes, up to isometry. Combining internal angle formulas with the Poincaré relation, we derive formulas for the Riemannian volume of these polytopes if the dimension $d$ is even. All formulas are stated in terms of the standard normal distribution function $Φ(x)$ and its imaginary version $Φ({\rm{i}} x)$. For example, if $d\geq 2$ is even, then the hyperbolic volume of the ideal regular simplex in the $d$-dimensional hyperbolic space of curvature $κ= -1$ is $$ \frac{π^{d/2}} {\sqrt{2}\, {\rm{i}}^{d}\, Γ\left(\frac{d+1}{2}\right)} \int_{-\infty}^{\infty} \left[ Φ\left(\frac{{\rm i} y}{\sqrt d}\right)^{d+1} + Φ\left(-\frac{{\rm i} y}{\sqrt d}\right)^{d+1} \right] {\rm e}^{-y^2/2} {\rm d} y. $$

math.PR↗
arXiv2026

Expected hyperbolic volumes of random beta polytopes

Let $X_1,\ldots,X_n$ be independent random points in the closed unit ball of $\mathbb{R}^d$. Assume that each $X_i$ has a beta distribution with parameter $β_i \ge -1$: if $β_i>-1$, then $X_i$ has Lebesgue density proportional to $(1-\|x\|^2)^{β_i}$ on $\{\|x\|<1\}$, whereas the case $β_i=-1$ corresponds to the uniform distribution on the unit sphere $\{\|x\|=1\}$. Let $[X_1,\ldots,X_n]$ denote the convex hull of these points. Interpreting the unit ball as the Klein model of hyperbolic geometry, we derive closed-form formulas for the expected hyperbolic volume of the random hyperbolic polytope $[X_1,\ldots,X_n]$. As a special case, if $X_1,\ldots,X_n$ are independent and uniformly distributed on the unit sphere in $\mathbb{R}^3$, then for every $n\ge 4$, \[ \mathbb{E}\,\operatorname{Vol}_{3}^{\mathrm{hyp}}\!\bigl([X_1,\ldots,X_n]\bigr) = π\left(\frac{n}{2}-\sum_{j=1}^{n-1}\frac{1}{j}\right). \]

math.PR↗
arXiv2025

Angles of orthocentric simplices

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.

math.MG↗