关于B.托特
On the diminishing process of B. Tóth
摘要：
令$K$和$K_0$是$\mathbb{R}^d$中的凸体，使得$K$包含原点，并定义过程$(K_n, p_n)$，$n \geq 0$如下：令$p_{n+1}$是$K_n$中的均匀随机点，并设置$K_{n+1} = K_n \cap (p_{n+1} + K)$。显然，$(K_n)$是一个凸体的嵌套序列，它收敛到一个非空的极限对象，也是$\mathbb{R}^d$中的凸体。我们研究这个过程中，$K$是一个规则的单纯形，立方体，或规则的凸多边形与奇数个顶点。对于一维非均匀分布，我们也得到了一些新的结果。
英文摘要：
Let $K$ and $K_0$ be convex bodies in $\mathbb{R}^d$, such that $K$ contains the origin, and define the process $(K_n, p_n)$, $n \geq 0$, as follows: let $p_{n+1}$ be a uniform random point in $K_n$, and set $K_{n+1} = K_n \cap (p_{n+1} + K)$. Clearly, $(K_n)$ is a nested sequence of convex bodies which converge to a non-empty limit object, again a convex body in $\mathbb{R}^d$. We study this process for $K$ being a regular simplex, a cube, or a regular convex polygon with an odd number of vertices. We also derive some new results in one dimension for non-uniform distributions.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
