集对函数的Choquet型扩张理论及其在图极限、超图、黎曼流形与度量测度空间中的应用
Choquet-type extensions of set-pair functions, and applications in graph limits, hypergraphs, and metric measure spaces
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中文总结 AI 辅助
本文提出集对函数的Choquet扩张等概念，构建集合分数阶优化的等价表示，推导测度空间的谱界，建立全局扩张常数的单调性不等式，为多类结构的相关几何与组合量提供新的界。
中文摘要 AI 辅助
我们提出集对函数的Choquet扩张、Choquet扩张的$L^p$积分以及全局扩张常数，并将这些概念用于研究优化问题并对诸多组合量与几何量进行上界估计。该研究方向也适用于原始Choquet扩张的研究，在该框架下得到了原始版本的平行结果。具体而言，我们利用Choquet型扩张构建了基于集合的分数阶优化的等价函数表示，该表示可应用于多种场景，如图极限或黎曼流形上的最大割、二分率与电导问题。我们进一步建立了一类Choquet扩张的$L^p$积分，并将其用于推导测度空间上电导率及其他组合量的谱界。我们提出了全局扩张常数的单调性不等式，该不等式统一了经典估计并为大量几何量与组合量揭示了新的界，包括扭转刚度、Cheeger常数、Dirichlet $p$-等周常数与$p$-拉普拉斯特征值，且这些界适用于完全不同的基础结构，涵盖超图、图极限、黎曼流形与度量测度空间。
英文摘要
We propose Choquet-type extensions for set-pair functions, an $L^p$-integration of a family of such extensions, and the associated min-max extension constants, and apply these to investigate variational problems and establish sharp bounds for many combinatorial and geometric quantities. Specifically, we use Choquet-type extensions to construct equivalent functional representations of fractional optimization problems with set-variables, which find applications in various settings, such as maxcut and bipartiteness ratio on graph limits, and Cheeger constants on Riemannian manifolds. We further establish the $L^p$ integration of a family of Choquet-type extensions, and apply it to derive spectral bounds for conductance and other combinatorial quantities on measure spaces. A monotonicity inequality on min-max extension constants is proposed, which unifies classical estimates and uncovers new bounds for many geometric and combinatorial quantities, such as torsional rigidity, Cheeger constants, Dirichlet $p$-isoperimetric constant, and $p$-Laplacian eigenvalues, in seemingly unrelated underlying structures----including hypergraphs, graph limits, Riemannian manifolds and metric measure spaces.
发表机构
- School of Mathematical Sciences, Peking University（北京大学数学科学学院）
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