A refinement of the Sylvester problem: Probabilities of combinatorial types

TL;DR

This paper refines the classical Sylvester problem by calculating probabilities of random convex hulls having specific combinatorial types in various distributions and geometric settings.

Contribution

It provides explicit probability formulas for different combinatorial types of convex hulls under multiple random distributions, extending classical results.

Findings

Computed probabilities for convex hulls of specific combinatorial types in normal, beta, and random walk distributions.

Recovered a recent solution to Youden's demon problem as a special case.

Solved the conic version of the Sylvester problem in several cases.

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 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 $\xi_1,\ldots, \xi_n$, the empirical mean $\frac 1n (\xi_1 + \ldots + \xi_n)$ 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.