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Lift zonoid order and functional inequalities

We introduce the notion of a weighted lift zonoid and show that, for properly chosen weights v, the ordering condition on a measure μ, 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 μ. We observe that such a choice may be useful for proving direct log-Sobolev inequality, as well, either in its weighted or classical forms.