The Sylvester question in $\mathbb{R}^d$: convex sets with a flat floor

Abstract

Pick $n$ independent and uniform random points $U_1,\ldots,U_n$ in a compact convex set $K$ of $\mathbb{R}^d$ with volume 1, and let $P^{(d)}_K(n)$ be the probability that these points are in convex position. The Sylvester conjecture in $\mathbb{R}^d$ is that $\min_K P^{(d)}_K(d+2)$ is achieved by the $d$-dimensional simplices $K$ (only). In this paper, we focus on a companion model, already studied in the $2d$ case, which we define in any dimension $d$: we say that $K$ has $F$ as a flat floor, if $F$ is a subset of $K$, contained in a hyperplan $P$, such that $K$ lies in one of the half-spaces defined by $P$. We define $Q_K^F(n)$ as the probability that $U_1,\cdots,U_n$ together with $F$ are in convex position (i.e., the $U_i$ are on the boundary of the convex hull ${\sf CH}(\{U_1,\cdots,U_n\}\cup F\})$). We prove that, for all fixed $F$, $K\mapsto Q_K^F(2)$ reaches its minimum on the "mountains" with floor $F$ (mountains are convex hull of $F$ union an additional vertex), while the maximum is not reached, but $K\mapsto Q_K^F(2)$ has values arbitrary close to 1. If the optimisation is done on the set of $K$ contained in $F\times[0,d]$ (the "subprism case"), then the minimum is also reached by the mountains, and the maximum by the "prism" $F\times[0,1]$. Since again, $Q_K^F{(2)}$ relies on the expected volume (of ${\sf CH}(\{V_1,V_2\}\cup F\})$), this result can be seen as a proof of the Sylvester problem in the floor case. In $2d$, where $F$ can essentially be the segment $[0,1],$ we give a general decomposition formula for $Q_K^F(n)$ so to compute several formulas and bounds for different $K$. In 3D, we give some bounds for $Q_K^F(n)$ for various floors $F$ and special cases of $K$.