AI 中文总结
该研究提出了描述$\boldsymbol{\text{R}}^d$任意子域上极小局部凹函数结构的理论，定义了极值集并证明其叶状性、被特定单纯形覆盖，还得出一维极值集接近自由边界的方式。
AI 中文摘要
我们提出了一个理论，描述任意$\boldsymbol{\text{R}}^d$子域上极小局部凹函数的结构。我们引入了极值集的概念，即函数在其上为仿射的集合，并证明极值集构成该子域的叶状结构。我们表明，极值集由顶点位于子域边界上的单纯形覆盖，且这些单纯形本身包含在极值集内。我们还证明，一维极值集仅能以切向方式接近自由边界，而非横向方式。
英文摘要
We propose a theory describing the structure of minimal locally concave functions on an arbitrary subdomain of $\mathbb{R}^d$. We introduce the notion of an extremal set, that is, a set on which the function is affine, and prove that extremal sets foliate the domain. We show that extremal sets are covered by simplices with vertices on the boundary of the domain, and that these simplices themselves lie in the extremal set. We also prove that one-dimensional extremal sets can approach the free boundary only tangentially, not transversally.
Journal refAlgebra i Analiz, Vol. 38 (2026), No. 5, pp. 33-95
