A refinement of the Sylvester problem: Probabilities of combinatorial types

TL;DR

The paper refines Sylvester's problem by classifying convex hulls of d+2 random points in R^d into combinatorial types T_m^d and deriving p_{d,m} for several distributions and models. It provides explicit, often integral, formulas linking p_{d,m} to the expected f-vector or to known combinatorial numbers (Eulerian, B-Eulerian) for Gaussian, beta-type, beta'-type points, and for random walks/bridges, including a spherical/cone analogue via Wendel–Donoho–Tanner theory. A key insight is the equivalence between refined Sylvester probabilities and Youden's demon problem in the Gaussian case, enabling recovery of recent results, and the results extend to conic/spherical variants with analogous structure. The work combines Radon partitions, cone angles, and Eulerian-number techniques to yield closed forms and integral representations that illuminate the geometry of random polytopes with minimal vertex counts and their asymptotic behavior. The findings have potential implications for stochastic geometry, high-dimensional probability, and related combinatorial geometry problems.

Abstract

Let $X_1,\ldots, X_{d+2}$ be random points in $\mathbb R^d$. The classical Sylvester problem asks to determine the probability that the convex hull of these points, denoted by $P:= [X_1,\ldots, X_{d+2}]$, is a simplex. In the present paper, we study a refined version of this problem which asks to determine the probability that $P$ has a given combinatorial type. It is known that there are $\lfloor d/2\rfloor+1$ possible combinatorial types of simplicial $d$-dimensional polytopes with at most $d+2$ vertices. These types are denoted by $T_0^d, T_1^d, \ldots, T_{\lfloor d/2 \rfloor}^d$, where $T_0^d$ is a simplex with $d+1$ vertices, while the remaining types have exactly $d+2$ vertices. Our aim is thus to compute the probability $$ p_{d,m} := \mathbb P[P \text{ is of type } T_{m}^d], \qquad m\in \{0,1,\ldots, \lfloor d/2 \rfloor\}. $$ The classical Sylvester problem corresponds to the case $m=0$. We shall compute $p_{d,m}$ for all $m$ in the following cases: (a) $X_1,\ldots, X_{d+2}$ are i.i.d. normal; (b) $X_1,\ldots, X_{d+2}$ follow a $d$-dimensional beta or beta prime distribution, which includes the uniform distribution on the ball or on the sphere as special cases; (c) $X_1,\ldots, X_{d+2}$ form a random walk with symmetrically exchangeable increments. As a by-product of case (a) we recover a recent solution to Youden's demon problem which asks to determine the probability that, in a one-dimensional i.i.d. normal sample $ξ_1,\ldots, ξ_n$, the empirical mean $\frac 1n (ξ_1 + \ldots + ξ_n)$ lies between the $k$-th and the $(k+1)$-st order statistics. We also consider the conic (or spherical) version of the refined Sylvester problem and solve it in several special cases.

Theorems & Definitions (50)

• Theorem 2.1: See Section 6.1 in gruenbaum_book
• Theorem 2.2: See Section 6.1 in gruenbaum_book
• Theorem 3.1: Probabilities of types and expected $f$-vector
• proof
• Theorem 3.2: Probabilities of types for Gaussian points
• Lemma 3.3
• proof
• proof : Proof of Theorem \ref{['theo:sylvester_probab_types_gauss']}
• Corollary 3.4: Sylvester problem for Gaussian points
• proof