arXiv 2102.076092021-02math.FA

关于低维集值映射的核

On the Core of a Low Dimensional Set-Valued Mapping

Pavel Shvartsman

备注：: 54 pages and 32 figures. This is the second version of the article containing detailed proofs of the main and auxiliary results. arXiv admin note: substantial text overlap with arXiv:2010.04540

摘要：

设${\mathfrak M}=({\mathcal M},ρ)$是度量空间，$X$是Banach空间.设$F$是从${\mathcal M}$到$X$的所有维数至多为$m$的紧凸子集族${\mathcal K}_m(X)$的集值映射.在我们最近与Charles Febriman的联合论文中的主要结果（被称为“Lipschitz选择的独立性原理”）提供了$F$的Lipschitz选择存在的有效条件，即，Lipschitz映射$f:{\mathcal M}\to X$，使得对于每个$x\in{\mathcal M}$都是$f(x)\in F(x)$。在两种特殊情况下，我们给出了这个结果的新的替代证明。当$m=2$时，我们为$X={\bf R}^{2}$证明它，当$m=1$时，我们为$X$的所有选择证明它。这两个证明都利用了一个简单的迭代公式来表示集值映射$F$的“核”，即，对于映射$G:{\mathcal M}\to{\mathcal K}_m(X)$，其是关于Hausdorff距离的Lipschitz，并且使得对于所有$x\in{\mathcal M}$的$G(x)\subset F(x)$。

英文摘要：

Let ${\mathfrak M}=({\mathcal M},ρ)$ be a metric space and let $X$ be a Banach space. Let $F$ be a set-valued mapping from ${\mathcal M}$ into the family ${\mathcal K}_m(X)$ of all compact convex subsets of $X$ of dimension at most $m$. The main result in our recent joint paper with Charles Fefferman (which is referred to as a "Finiteness Principle for Lipschitz selections") provides efficient conditions for the existence of a Lipschitz selection of $F$, i.e., a Lipschitz mapping $f:{\mathcal M}\to X$ such that $f(x)\in F(x)$ for every $x\in{\mathcal M}$. We give new alternative proofs of this result in two special cases. When $m=2$ we prove it for $X={\bf R}^{2}$, and when $m=1$ we prove it for all choices of $X$. Both of these proofs make use of a simple reiteration formula for the "core" of a set-valued mapping $F$, i.e., for a mapping $G:{\mathcal M}\to{\mathcal K}_m(X)$ which is Lipschitz with respect to the Hausdorff distance, and such that $G(x)\subset F(x)$ for all $x\in{\mathcal M}$.

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。

摘要：

英文摘要：

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