Functional Covering Numbers

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, b...

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Bibliographic Details
Published inThe Journal of geometric analysis Vol. 31; no. 1; pp. 1039 - 1072
Main Authors Artstein-Avidan, Shiri, Slomka, Boaz A.
Format Journal Article
LanguageEnglish
Published New York Springer US 2021
Springer Nature B.V
Subjects
Abstract Harmonic Analysis
Convex and Discrete Geometry
Differential Geometry
Dynamical Systems and Ergodic Theory
Fourier Analysis
Functionals
Geometry
Global Analysis and Analysis on Manifolds
Inequalities
Mathematics
Mathematics and Statistics
Separation
Covering numbers
Log-concave functions
52A23
Functionalization of geometry
Duality
position
46A20
Volume bounds
52C17
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Summary: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.
ISSN:1050-6926
1559-002X
DOI:10.1007/s12220-019-00310-3