论文标题
凸体中随机点的平均距离的急剧不平等
Sharp inequalities for the mean distance of random points in convex bodies
论文作者
论文摘要
对于凸面，$ k \ subset \ mathbb {r}^d $平均距离$δ（k）= \ mathbb {e} | x_1-x_2 | $是两个独立和均匀分布的随机点的欧几里得距离$ x_1，x_1，x_1，x_2，x_2 \ in k $ in k $。得出了$Δ（k）$和第一个内在体积$ v_1（k）$ $ k $（归一化平均宽度）的最佳上限和上限。该论点依赖Riesz的重排不平等，以及解决凹入功能的优化问题的解决方案。详细回顾了与现有文献所知的结果的关系。
For a convex body $K\subset\mathbb{R}^d$ the mean distance $Δ(K)=\mathbb{E}|X_1-X_2|$ is the expected Euclidean distance of two independent and uniformly distributed random points $X_1,X_2\in K$. Optimal lower and upper bounds for ratio between $Δ(K)$ and the first intrinsic volume $V_1(K)$ of $K$ (normalized mean width) are derived and degenerate extremal cases are discussed. The argument relies on Riesz's rearrangement inequality and the solution of an optimization problem for powers of concave functions. The relation with results known from the existing literature is reviewed in detail.
