论文标题
与Weyl腔有关的锥形镶嵌物
Conical tessellations associated with Weyl chambers
论文作者
论文摘要
我们考虑$ d $ dimensional随机向量$ y_1，\ dots，y_n $满足温和的一般位置假设A.S.超平面\ begin {align*}（y_i-y_j）^\ perp \; \; （1 \ le i <j \ le n）。 \ end {align*}生成欧几里得$ d $ -space的圆锥形镶嵌，该孔与$ a_ {n-1} $类型的Weyl Charbers密切相关。我们确定锥形中的圆锥体数，并表明它是A.S.持续的。对于从这种随机镶嵌中随机选择的随机锥体，我们计算对一系列几何函数的期望。这些包括面部数字以及圆锥固有体积和圆锥形Quermassintegrals。在$ y_1，\ ldots，y_n $上交换的其他假设，对双随机锥的分布相同，该双随机锥具有与$ y_1-y_2，\ ldots，y_ __ {n-1} -y_n $相同的正壳的分布相同的分布。所有这些期望都没有发行。同样，我们考虑由超平面\ begin {align*}（y_i+y_j）^\ perp \; \; \;;; （1 \ le i <j \ le n），\ quad（y_i-y_j）^\ perp \; \; \; （1 \ le i <j \ le n），\ quad y_i^\ perp \; \; \; （1 \ le i \ le n）\ end {align*}此镶嵌与$ b_n $的韦尔室密切相关。我们计算此镶嵌物中的锥量数，以及从这种随机镶嵌中得出的随机锥体的各种几何函数的期望。证明中的主要成分是镶嵌面的面部数量与相应类型的Weyl室的面部数量之间的连接，这些面部由一般位置的某个线性子空间相交。
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.
