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Orlicz Addition for Measures and an Optimization Problem for the $f$-divergence

Part of: General convexity Foundations of probability theory Classical measure theory Functions of several variables Communication, information

Published online by Cambridge University Press:  16 July 2019

Shaoxiong Hou  and
Deping Ye
Shaoxiong Hou
Affiliation:
College of Mathematics and Information Science, Hebei Normal University, Shijiazhuang, Hebei Province, 050024, P. R. China Department of Mathematics and Statistics, Memorial University, St. John’s, NL A1C 5S7 Email: s.hou@mun.cadeping.ye@mun.ca
Deping Ye
Affiliation:
College of Mathematics and Information Science, Hebei Normal University, Shijiazhuang, Hebei Province, 050024, P. R. China Department of Mathematics and Statistics, Memorial University, St. John’s, NL A1C 5S7 Email: s.hou@mun.cadeping.ye@mun.ca
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Abstract

This paper provides a functional analogue of the recently initiated dual Orlicz–Brunn–Minkowski theory for star bodies. We first propose the Orlicz addition of measures, and establish the dual functional Orlicz–Brunn–Minkowski inequality. Based on a family of linear Orlicz additions of two measures, we provide an interpretation for the famous $f$-divergence. Jensen’s inequality for integrals is also proved to be equivalent to the newly established dual functional Orlicz–Brunn–Minkowski inequality. An optimization problem for the $f$-divergence is proposed, and related functional affine isoperimetric inequalities are established.

Keywords

affine isoperimetric inequalitydual Brunn–Minkowski theoryf-divergenceoptimization problem for the f-divergence

MSC classification

Primary: 52A41: Convex functions and convex programs
Secondary: 26B25: Convexity, generalizations 28A25: Integration with respect to measures and other set functions 60A10: Probabilistic measure theory 94A15: Information theory, general 94A17: Measures of information, entropy
Type
Article
Information
Canadian Journal of Mathematics , Volume 72 , Issue 2 , April 2020 , pp. 455 - 479
Copyright
© Canadian Mathematical Society 2019

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Footnotes

The research of S. H. has been partially supported by CSC (No. 201306040135) and by the Science Foundation of Hebei Normal University (No. L2018B32). The research of D. Y. is supported by a NSERC grant.

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