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Minimal subfamilies and the probabilistic interpretation for modulus on graphs

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Abstract

The notion of p-modulus of a family of objects on a graph is a measure of the richness of such families. We develop the notion of minimal subfamilies using the method of Lagrangian duality for p-modulus. We show that minimal subfamilies have at most |E| elements and that these elements carry a weight related to their “importance” in relation to the corresponding p-modulus problem. When \(p=2\), this measure of importance is in fact a probability measure and modulus can be thought as trying to minimize the expected overlap in the family.

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Acknowledgments

Research partially supported by NSF Grants DMS-1201427 and DMS-1515810.

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Authors and Affiliations

  1. Department of Mathematics, Kansas State University, Manhattan, KS, 66506, USA

    Nathan Albin & Pietro Poggi-Corradini

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  1. Nathan Albin
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  2. Pietro Poggi-Corradini
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Correspondence to Pietro Poggi-Corradini.

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On the occasion of David Minda’s retirement.

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Albin, N., Poggi-Corradini, P. Minimal subfamilies and the probabilistic interpretation for modulus on graphs. J Anal 24, 183–208 (2016). https://doi.org/10.1007/s41478-016-0002-9

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  • DOI: https://doi.org/10.1007/s41478-016-0002-9

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