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

Abstract Consider a finite collection of affine hyperplanes in $$\mathbb R^d$$ R d . The hyperplanes dissect $$\mathbb R^d$$ R d into finitely many polyhedral chambers. For a point $$x\in \mathbb R^d$$ x ∈ R d and a chamber P the metric projection of x onto P is the unique point $$y\in P$$ y ∈ P minimizing the Euclidean distance to x. The metric projection is contained in the relative interior of a uniquely defined face of P whose dimension is denoted by $$\text {dim}(x,P)$$ dim ( x , P ) . We prove that for every given $$k\in \{0,\ldots , d\}$$ k ∈ { 0 , … , d } , the number of chambers P for which $$\text {dim}(x,P) = k$$ dim ( x , P ) = k does not depend on the choice of x, with an exception of some Lebesgue null set. Moreover, this number is equal to the absolute value of the k-th coefficient of the characteristic polynomial of the hyperplane arrangement. In a special case of reflection arrangements, this proves a conjecture of Drton and Klivans [A geometric interpretation of the characteristic polynomial of reflection arrangements. Proc. Amer. Math. Soc. 138(8), 2873–2887 (2010)].

Abstract We consider the simplices $$\begin{aligned} K_n^A=\{x\in {\mathbb {R}}^{n+1}:x_1\ge x_2\ge \cdots \ge x_{n+1},x_1-x_{n+1}\le 1,\,x_1+\cdots +x_{n+1}=0\} \end{aligned}$$ K n A = { x ∈ R n + 1 : x 1 ≥ x 2 ≥ ⋯ ≥ x n + 1 , x 1 - x n + 1 ≤ 1 , x 1 + ⋯ + x n + 1 = 0 } and $$\begin{aligned} K_n^B=\{x\in {\mathbb {R}}^n:1\ge x_1\ge x_2\ge \cdots \ge x_n\ge 0\}, \end{aligned}$$ K n B = { x ∈ R n : 1 ≥ x 1 ≥ x 2 ≥ ⋯ ≥ x n ≥ 0 } , which are called the Schläfli orthoschemes of types A and B, respectively. We describe the tangent cones at their j-faces and compute explicitly the sums of the conic intrinsic volumes of these tangent cones at all j-faces of $$K_n^A$$ K n A and $$K_n^B$$ K n B . This setting contains sums of external and internal angles of $$K_n^A$$ K n A and $$K_n^B$$ K n B as special cases. The sums are evaluated in terms of Stirling numbers of both kinds. We generalize these results to finite products of Schläfli orthoschemes of type A and B and, as a probabilistic consequence, derive formulas for the expected number of j-faces of the Minkowski sums of the convex hulls of a finite number of Gaussian random walks and random bridges. Furthermore, we evaluate the analogous angle sums for the tangent cones of Weyl chambers of types A and B and finite products thereof.

Abstract Consider a random simplex $$[X_1,\ldots ,X_n]$$ [ X 1 , … , X n ] defined as the convex hull of independent identically distributed (i.i.d.) random points $$X_1,\ldots ,X_n$$ X 1 , … , X n in $$\mathbb {R}^{n-1}$$ R n - 1 with the following beta density: "Equation missing" Let $$J_{n,k}(\beta )$$ J n , k ( β ) be the expected internal angle of the simplex $$[X_1,\ldots ,X_n]$$ [ X 1 , … , X n ] at its face $$[X_1,\ldots ,X_k]$$ [ X 1 , … , X k ] . Define $${\tilde{J}}_{n,k}(\beta )$$ J ~ n , k ( β ) analogously for i.i.d. random points distributed according to the beta $$'$$ ′ density $${\tilde{f}}_{n-1,\beta } (x) \propto (1+\Vert x\Vert ^2)^{-\beta }, x\in \mathbb {R}^{n-1}, \beta > ({n-1})/{2}.$$ f ~ n - 1 , β ( x ) ∝ ( 1 + ‖ x ‖ 2 ) - β , x ∈ R n - 1 , β > ( n - 1 ) / 2 . We derive formulae for $$J_{n,k}(\beta )$$ J n , k ( β ) and $${\tilde{J}}_{n,k}(\beta )$$ J ~

Abstract The typical cell of a Voronoi tessellation generated by $$n+1$$ n + 1 uniformly distributed random points on the d-dimensional unit sphere $$\mathbb {S}^d$$ S d is studied. Its f-vector is identified in distribution with the f-vector of a beta’ polytope generated by n random points in $$\mathbb {R}^d$$ R d . Explicit formulas for the expected f-vector are provided for any d and the low-dimensional cases $$d\in \{2,3,4\}$$ d ∈ { 2 , 3 , 4 } are studied separately. This implies an explicit formula for the total number of k-dimensional faces in the spherical Voronoi tessellation as well.