Lift zonoid order and functional inequalities

TL;DR

This paper introduces weighted lift zonoids to establish new functional inequalities for measures, including extensions of Bobkov's shift inequality and a novel inverse log-Sobolev inequality, with potential applications to classical log-Sobolev inequalities.

Contribution

It defines weighted lift zonoids and demonstrates their use in deriving new functional inequalities, notably with a novel choice of weights for inverse log-Sobolev inequalities.

Findings

Derived non-linear extensions of Bobkov's shift inequality.

Introduced a new weighted inverse log-Sobolev inequality.

Suggested potential for proving classical log-Sobolev inequalities.

Abstract

We introduce the notion of a weighted lift zonoid and show that, for properly chosen weights v, the ordering condition on a measure \mu, formulated in terms of the weighted lift zonoids of this measure, leads to certain functional inequalities for this measure, such as non-linear extensions of Bobkov's shift inequality and weighted inverse log-Sobolev inequality. The choice of the weight K, involved in our version of the inverse log-Sobolev inequality, differs substantially from those available in the literature, and requires the weight v, involved into the definition of the weighted lift zonoid, to equal the divergence of the weight K w.r.t. initial measure \mu. We observe that such a choice may be useful for proving direct log-Sobolev inequality, as well, either in its weighted or classical forms.