Choquet-type extension theory of set-pair functions, and applications to graph limits, hypergraphs, Riemannian manifolds and metric measure spaces

Abstract

We propose Choquet extension for set-pair functions, $L^p$ integration of Choquet extensions, and global extension constants, and apply these to investigate optimization problems and bound many combinatorial and geometric quantities. Our research line is also applicable to the study of original Choquet extension; within this framework, parallel results for the original version are obtained. Specifically, we use Choquet-type extensions to build an equivalent functional representation of set-based fractional optimization, which finds applications in various settings, such as maxcut, bipartiteness ratio, and conductance on graph limits or Riemannian manifolds. We further establish the $L^p$ integration of a family of Choquet extensions, and apply it to derive spectral bounds for conductance and other combinatorial quantities on measure spaces. A monotonicity inequality on global extension constants is proposed, which unifies classical estimates and uncovers new bounds for a lot of geometric and combinatorial quantities, such as torsional rigidity, Cheeger constants, Dirichlet $p$-isoperimetric constant, and $p$-Laplacian eigenvalues, in totally distinct underlying structures----including hypergraphs, graph limits, Riemannian manifolds and metric measure spaces.