Year: 2026
Type: article
Abstract: Let $$X_1,\ldots , X_{d+2}$$ X 1 , … , X d + 2 be random points in $$\mathbb {R}^d$$ R d . The classical Sylvester problem asks to determine the probability that the convex hull of these points, denoted by $$P:= [X_1,\ldots , X_{d+2}]$$ P : = [ X 1 , … , 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 $$\lfloor d/2\rfloor +1$$ ⌊ d / 2 ⌋ + 1 possible combinatorial types of simplicial d -dimensional polytopes with at most $$d+2$$ d + 2 vertices. These types are denoted by $$T_0^d, T_1^d, \ldots , T_{\lfloor d/2 \rfloor }^d$$ T 0 d , T 1 d , … , T ⌊ d / 2 ⌋ d , where $$T_0^d$$ T 0 d is a simplex with $$d+1$$ d + 1 vertices, while the remaining types have exactly $$d+2$$ d + 2 vertices. Our aim is thus to compute the probability $$ p_{d,m} := \mathbb {P}[P \text { is of type } T_{m}^d], \qquad m\in \{0,1,\ldots , \lfloor d/2 \rfloor \}. $$ p d , m : = P [ P is of type
Authors: Zakhar Kabluchko, Hugo Panzo
Institutions: University of Münster, Saint Louis University
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: hybrid
Estimated APC paid: $3,090
Funder: Deutsche Forschungsgemeinschaft
OpenAlex ID: W7128719474