Geodesic
Orlicz Addition for Measures and an Optimization Problem for the $f$-divergence
Canadian journal of mathematics, Tome 72 (2020) no. 2, pp. 455-479

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DOI

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.
DOI : 10.4153/S0008414X19000117
Mots-clés : affine isoperimetric inequality, dual Brunn–Minkowski theory, f-divergence, optimization problem for the f-divergence 
Hou, Shaoxiong; Ye, Deping. Orlicz Addition for Measures and an Optimization Problem for the $f$-divergence. Canadian journal of mathematics, Tome 72 (2020) no. 2, pp. 455-479. doi: 10.4153/S0008414X19000117
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     title = {Orlicz {Addition} for {Measures} and an {Optimization} {Problem} for the $f$-divergence},
     journal = {Canadian journal of mathematics},
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     year = {2020},
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