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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中文翻译：

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与外尔室相关的锥形镶嵌

摘要：我们考虑 $d$ 维随机向量 $Y_1,\ldots ,Y_n$ 满足一个温和的一般位置假设作为超平面 \begin{align*} (Y_i-Y_j)^\perp \;\; (1\le i<j\le n) \end{align*} 生成欧几里得 $d$-空间的锥形镶嵌，这与 $A_{n-1}$ 类型的 Weyl 室密切相关。我们确定了这个细分中的锥体数量，并表明它是恒定的。对于从这个随机细分中随机均匀选择的随机锥，我们计算了几个几何函数的期望。这些包括面数，以及圆锥固有体积和圆锥 quermassintegrals。在 $Y_1,\ldots ,Y_n$ 上的可交换性的附加假设下，对于与 $Y_1-Y_2,\ldots 的正壳具有相同分布的双随机锥也是如此，Y_{n-1}-Y_n$ 以该正壳不等于 $\mathbb R^d$ 为条件。所有这些期望都证明是无分配的。类似地，我们考虑由超平面 \begin{align*} (Y_i+Y_j)^\perp \;\; 引起的锥形镶嵌。(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*} 这种镶嵌与 $B_n$ 类型的 Weyl 室密切相关。我们计算该镶嵌中的锥体数量以及从该随机镶嵌中提取的随机锥的各种几何函数的期望。

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