论文标题
二维设置值映射的核心。 Lipschitz选择低维设置值映射的Lipschitz选择的存在标准和有效算法
The Core of a 2-Dimensional Set-Valued Mapping. Existence Criteria and Efficient Algorithms for Lipschitz Selections of Low Dimensional Set-Valued Mappings
论文作者
论文摘要
令$ {\ Mathfrak m} =（{\ Mathcal m}，ρ）$为公制空间，让$ x $为Banach空间。令$ f $是从$ {\ mathcal m} $中的设置值映射到家庭$ {\ mathcal k} _m（x）$（x）$的所有紧凑型凸subset的$ x $，最多是$ m $的$ x $。我们最近与Charles Fefferman（称为“ Lipschitz选择的有限原理”）的联合论文的主要结果为存在$ f $的存在提供了有效条件M} $。在两种特殊情况下，我们提供了有关此结果的新替代证明。当$ m = 2 $时，我们以$ x = {\ bf r}^{2} $证明它，当$ m = 1 $时，我们以$ x $的所有选择来证明它。这两种证据都利用了一个简单的重申公式，用于设置值映射$ f $的“核心”，即，对于映射$ g：{\ Mathcal m} \ to {\ Mathcal k} _m（x）$ at lipschitz at lipsChitz，对Hausdorff decanc. $ x \ in {\ Mathcal M} $。我们还提出了几个建设性标准，即Lipschitz在$ {\ Mathcal m} $中选择了set值映射的选择，以$ {\ bf r}^{2} $中的所有封闭的半平台的家族。
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 $x\in{\mathcal M}$. We also present several constructive criteria for the existence of Lipschitz selections of set-valued mappings from ${\mathcal M}$ into the family of all closed half-planes in ${\bf R}^{2}$.
