Conical tessellations associated with Weyl chambers

Abstract

We consider $d$-dimensional random vectors $Y_1,\dots,Y_n$ that satisfy a mild general position assumption a.s. The hyperplanes \begin{align*} (Y_i-Y_j)^\perp\;\; (1\le i<j\le n). \end{align*} generate a conical tessellation of the Euclidean $d$-space which is closely related to the Weyl chambers of type $A_{n-1}$. We determine the number of cones in this tessellation and show that it is a.s. constant. For a random cone chosen uniformly at random from this random tessellation, we compute expectations for a general series of geometric functionals. These include the face numbers, as well as the conic intrinsic volumes and the conic quermassintegrals. Under the additional assumption of exchangeability on $Y_1,\ldots,Y_n$, the same is done for the dual random cones which have the same distribution as the positive hull of $Y_1-Y_2,\ldots, Y_{n-1}-Y_n$ conditioned on the event that this positive hull is not equal to $\mathbb R^d$. All these expectations turn out to be distribution-free. Similarly, we consider the conical tessellation induced by the hyperplanes \begin{align*} (Y_i+Y_j)^\perp\;\; (1 \le i<j\le n),\quad (Y_i-Y_j)^\perp\;\; (1\le i<j\le n),\quad Y_i^\perp\;\; (1\le i\le n) \end{align*} This tessellation is closely related to the Weyl chambers of type $B_n$. We compute the number of cones in this tessellation and the expectations of various geometric functionals for random cones drawn from this random tessellation. The main ingredient in the proofs is a connection between the number of faces of the tessellation and the number of faces of the Weyl chambers of the corresponding type that are intersected by a certain linear subspace in general position.