Orlicz Addition for Measures and an Optimization Problem for the -divergence
Keyword(s):
AbstractThis 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.
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2011 ◽
Vol 54
(9-10)
◽
pp. 2451-2459
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2013 ◽
Vol 4
(2)
◽
pp. 183-194
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