f-Divergence for convex bodies

2012/05/15 by Elisabeth M. Werner, Werner, Elisabeth M.
Computer Science · Mathematics · #52A20 #53A15 #FOS: Computer and information sciences #FOS: Mathematics #Functional Analysis (math.FA) #Information Theory (cs.IT) #cs.IT #math.FA #math.IT #msc:52A20 #msc:53A15

paper · pdf · doi:10.48550/arxiv.1205.3423

arxiv created 2012/05/15 · arxiv updated 2012/05/16

Abstract

We introduce f-divergence, a concept from information theory and statistics, for convex bodies in Rⁿ. We prove that f-divergences are SL(n) invariant valuations and we establish an affine isoperimetric inequality for these quantities. We show that generalized affine surface area and in particular the L_p affine surface area from the L_p Brunn Minkowski theory are special cases of f-divergences.

Related

• Rényi Divergence and L_p-affine surface area for convex bodies
• Curvature functionals on convex bodies
• Functional versions of L_p-affine surface area and entropy inequalities
• Pinsker inequalities and related Monge-Ampère equations for log concave functions
• Mixed f-divergence and inequalities for log concave functions