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Sobolev inequalities for probability measures on the real line

F. Barthe;  C. Roberto — 2003

Studia Mathematica

We give a characterization of those probability measures on the real line which satisfy certain Sobolev inequalities. Our starting point is a simpler approach to the Bobkov-Götze characterization of measures satisfying a logarithmic Sobolev inequality. As an application of the criterion we present a soft proof of the Latała-Oleszkiewicz inequality for exponential measures, and describe the measures on the line which have the same property. New concentration inequalities for product measures follow....

Shift inequalities of Gaussian type and norms of barycentres

F. Barthe;  D. Cordero-Erausquin;  M. Fradelizi — 2001

Studia Mathematica

We derive the equivalence of different forms of Gaussian type shift inequalities. This completes previous results by Bobkov. Our argument strongly relies on the Gaussian model for which we give a geometric approach in terms of norms of barycentres. Similar inequalities hold in the discrete setting; they improve the known results on the so-called isodiametral problem for the discrete cube. The study of norms of barycentres for subsets of convex bodies completes the exposition.

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