Functional Covering Numbers

  • Shiri Artstein-AvidanEmail author
  • Boaz A. Slomka


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.


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

Mathematics Subject Classification

52C17 52A23 46A20 



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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Copyright information

© Mathematica Josephina, Inc. 2019

Authors and Affiliations

  1. 1.School of Mathematical SciencesTel Aviv UniversityTel AvivIsrael
  2. 2.Department of Mathematics and Computer ScienceThe Open University of IsraelRa’ananaIsrael

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