Lipschitz映射强最小值可达性的双Lipschitz刻画
A bi-Lipschitz characterization of strong minimum-attainment for Lipschitz maps
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- Department of Mathematics Education, Sunchon National University（顺天国立大学数学教育系）
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中文总结 AI 辅助
本文用双Lipschitz嵌入完全刻画了强最小值可达Lipschitz函数稠密性的失败条件，并给出纯1-不可修正性的等距刻画，同时展示向量值情形的反例。
中文摘要 AI 辅助
我们完全刻画了强最小值可达Lipschitz函数（强范数可达Lipschitz函数的最小值类比）的稠密性，其刻画条件用双Lipschitz嵌入表示。更精确地，我们的主要结果表明，定义在完备度量空间$M$上的强最小值可达Lipschitz函数集合不稠密当且仅当$M$双Lipschitz等价于$\mathbb{R}$中具有正Lebesgue测度的子集，或等价地，当$M$允许双Lipschitz嵌入到$\mathbb{R}$且$M$具有正的一维Hausdorff测度。作为推论，我们给出了$M$的纯1-不可修正性的等距刻画，该刻画用定义在$M$的闭子集的双Lipschitz副本上的强最小值可达Lipschitz映射表示。我们还提供了几个反例，表明主要结果不能自然地推广到向量值情形。
英文摘要
We completely characterize the denseness of strongly minimum-attaining Lipschitz functions, a minimum analogue for strongly norm-attaining Lipschitz functions, in terms of bi-Lipschitz embeddings. More precisely, our main result shows that the set of strongly minimum-attaining Lipschitz functions defined on a complete metric space $M$ fails the denseness if and only if $M$ is bi-Lipschitz equivalent to a subset of $\mathbb{R}$ with positive Lebesgue measure, or equivalently, if $M$ admits a bi-Lipschitz embedding into $\mathbb{R}$ and $M$ has positive 1-dimensional Hausdorff measure. As a consequence, we provide an isometric characterization of the pure 1-unrectifiability of $M$ in terms of strongly minimum-attaining Lipschitz maps defined on bi-Lipschitz copies of closed subsets of $M$. Several counterexamples showing that the main result cannot be naturally extended to the vector-valued setting are also presented.
