Functional Covering Numbers

Abstract

We define covering and separation numbers for functions. We investigate their properties, and show that for some classes of functions there is exact equality between separation and covering numbers. We provide analogues for various geometric inequalities on covering numbers, such as volume bounds, bounds connected with Hadwiger’s conjecture, and inequalities about M-positions for geometric log-concave functions. In particular we get strong versions of M-positions for geometric log-concave functions.

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Acknowledgements

We thank Daniel Rosen for his valuable remarks for fruitful discussions. We also thank the anonymous referees for helpful remarks. This publication is a part of a project that has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation program (Grant Agreement No. 770127). The first named author was supported by ISF Grant Number 665/15.

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Correspondence to Shiri Artstein-Avidan.

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Artstein-Avidan, S., Slomka, B.A. Functional Covering Numbers. J Geom Anal 31, 1039–1072 (2021). https://doi.org/10.1007/s12220-019-00310-3

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Keywords

  • Covering numbers
  • Functionalization of geometry
  • Log-concave functions
  • Duality
  • Volume bounds
  • M-position

Mathematics Subject Classification

  • 52C17
  • 52A23
  • 46A20