论文标题
随机部分和随机单纯性不等式
Random section and random simplex inequality
论文作者
论文摘要
考虑一些凸面$ k \ subset \ mathbb r^d $。令$ x_1，\ dots，x_k $，其中$ k \ leq d $是独立且均匀地选择$ k $的随机点，让$ξ_k$是均匀分布的随机线性$ k $平面。我们表明，对于$ p \ geq-d+k+1 $，\ [\ mathbb e \，| k \capξ_k|^{d+p} \ leq c_ {d，d，k，k，p} \ cdot | k |^k |^k \ \，\，\，\，\，\ mathb e \，| \]其中$ | \ cdot | $和$ \ mathrm {cons} $表示通讯尺寸和凸形船体的卷。常数$ c_ {d，k，p} $是这样的，对于$ k> 1 $，当时且仅当$ k $是以原点为中心的椭圆形，而对于$ k = 1 $，不平等将不等式变成平等。如果$ p = 0 $，则不等式将减少到Busemann交叉点不等式，如果$ k = D $ - 到Busemann Random Simplex不等式。我们还提出了这种不平等现象的仿射版本，该版本类似地概括了施耐德不平等和Blaschke-Grömer不平等。
Consider some convex body $K\subset\mathbb R^d$. Let $X_1,\dots, X_k$, where $k\leq d$, be random points independently and uniformly chosen in $K$, and let $ξ_k$ be a uniformly distributed random linear $k$-plane. We show that for $p\geq-d+k+1$, \[ \mathbb E\,|K\capξ_k|^{d+p}\leq c_{d,k,p} \cdot|K|^k\, \,\mathbb E\,|\mathrm{conv}(0,X_1, \dots,X_k)|^p, \] where $|\cdot|$ and $\mathrm{conv}$ denote the volume of correspondent dimension and the convex hull. The constant $c_{d,k,p}$ is such that for $k>1$ the equality holds if and only if $K$ is an ellipsoid centered at the origin, and for $k=1$ the inequality turns to equality. If $p=0$, then the inequality reduces to the Busemann intersection inequality, and if $k=d$ -- to the Busemann random simplex inequality. We also present an affine version of this inequality which similarly generalizes the Schneider inequality and the Blaschke-Grömer inequality.
