论文标题
椭圆形的内在体积
Intrinsic volumes of ellipsoids
论文作者
论文摘要
我们根据椭圆形积分来推导$ \ Mathbb r^d $，$ d \ ge 2 $的椭圆形固有卷的显式公式。也就是说，对于椭圆形$ {\ Mathcal E} \ subset \ mathbb r^d $带有semiaxes $ a_1，\ ldots，a_d $，我们显示 \ begin {align*} v_k（{\ Mathcal e}）=κ_k\ sum_ {i = 1}^da_i^2s_ {k-1}（a_1^2，\ dots，a_ {i-1}^2，a_ {i+1}^2，\ dots，\ dots，\ dots，a_d^2）\ i nt_0^{\ infty} {t^{k-1} \ over（a_i^2t^2+1）\ prod_ {j = 1}^d \ sqrt {a_j^2t^2+1}}}}} \，\ rm {d} t \ end {align*}对于所有$ k = 1，\ ldots，d $，其中$ s_ {k-1} $是$（k-1）$ - 基本对称的多项式和$κ_k$是$ k $ dimensional单位球的体积。在我们的配方看起来特别简单的情况下，给出了一些内在卷$ v_k $的示例。作为一个应用程序，我们为预期的$ k $二维量的随机$ k $ -simplex提供了新的公式。
We deduce explicit formulae for the intrinsic volumes of an ellipsoid in $\mathbb R^d$, $d\ge 2$, in terms of elliptic integrals. Namely, for an ellipsoid ${\mathcal E}\subset \mathbb R^d$ with semiaxes $a_1,\ldots, a_d$ we show that \begin{align*} V_k({\mathcal E})=κ_k\sum_{i=1}^da_i^2s_{k-1}(a_1^2,\dots,a_{i-1}^2,a_{i+1}^2,\dots,a_d^2)\int_0^{\infty}{t^{k-1}\over(a_i^2t^2+1)\prod_{j=1}^d\sqrt{a_j^2t^2+1}}\,\rm{d}t \end{align*} for all $k=1,\ldots,d$, where $s_{k-1}$ is the $(k-1)$-th elementary symmetric polynomial and $κ_k$ is the volume of the $k$-dimensional unit ball. Some examples of the intrinsic volumes $V_k$ with low and high $k$ are given where our formulae look particularly simple. As an application we derive new formulae for the expected $k$-dimensional volume of random $k$-simplex in an ellipsoid and random Gaussian $k$-simplex.
