On the mixed $f$-divergence for multiple pairs of measures

TL;DR

This paper introduces the mixed $f$-divergence, extending classical divergence measures to multiple pairs of measures, establishing key properties, inequalities, and applications in convex body theory.

Contribution

It proposes the mixed $f$-divergence for multiple measure pairs, extending classical divergence concepts with new properties and inequalities.

Findings

Established permutation invariance and symmetry properties.

Proved Alexandrov-Fenchel type inequality.

Applied results to convex body theory.

Abstract

In this paper, the concept of the classical $f$-divergence (for a pair of measures) is extended to the mixed $f$-divergence (for multiple pairs of measures). The mixed $f$-divergence provides a way to measure the difference between multiple pairs of (probability) measures. Properties for the mixed $f$-divergence are established, such as permutation invariance and symmetry in distributions. An Alexandrov-Fenchel type inequality and an isoperimetric type inequality for the mixed $f$-divergence will be proved and applications in the theory of convex bodies are given.