Conical tessellations associated with Weyl chambers

Godland, Thomas; Kabluchko, Zakhar

doi:10.1090/tran/8445

Conical tessellations associated with Weyl chambers
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by Thomas Godland and Zakhar Kabluchko PDF

Trans. Amer. Math. Soc. 374 (2021), 7161-7196 Request permission

Abstract:

We consider $d$-dimensional random vectors $Y_1,\ldots ,Y_n$ 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 of several geometric functionals. These include the face numbers, as well as the conic intrinsic volumes and the conical 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 non-trivially by a certain linear subspace in general position.

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