Choquet-type extensions of set-pair functions, and applications in graph limits, hypergraphs, and metric measure spaces
Abstract
We propose Choquet-type extensions for set-pair functions, an Lp-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 Lp 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.
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BibTeX
@misc{zhang2026choquettype,
title = {{Choquet-type extensions of set-pair functions, and applications in graph limits, hypergraphs, and metric measure spaces}},
author = {Dong Zhang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2608.09606},
eprint = {2608.09606},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.09606},
}
RIS
TY - UNPB
TI - Choquet-type extensions of set-pair functions, and applications in graph limits, hypergraphs, and metric measure spaces
AU - Dong Zhang
PY - 2026
DA - 2026/10/01
PB - arXiv
DO - 10.48550/arxiv.2608.09606
UR - https://arxiv.org/abs/2608.09606
AB - 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.
ER -