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Lift zonoid order and functional inequalities

Authors: Alexei M. Kulik and Taras D. Tymoshkevych
Original publication: Teoriya Imovirnostei ta Matematichna Statistika, tom 89 (2013).
Journal: Theor. Probability and Math. Statist. 89 (2014), 83-99
MSC (2010): Primary 26D10, 39B62, 47D07, 60E15, 60J60
Published electronically: January 26, 2015
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Abstract: We introduce the notion of a weighted lift zonoid, and we show that the ordering condition imposed on a measure $ \mu $ and formulated in terms of the weighted lift zonoids of this measure leads to certain functional inequalities for $ \mu $ 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 known in the literature, and it requires that the weight $ v$ involved in the definition of the weighted lift zonoid equals the divergence of the weight $ K$ with respect to the initial measure $ \mu $. We show that such a choice may be useful for proving the direct log-Sobolev inequality as well.

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Additional Information

Alexei M. Kulik
Affiliation: Institute of Mathematics, National Academy of Science of Ukraine, Tereshchenkivska Street 3, 01601, Kyiv, Ukraine

Taras D. Tymoshkevych
Affiliation: National Taras Shevchenko University of Kyiv, Academician Glushkov Avenue 4-e, 03127, Kyiv, Ukraine

Keywords: Lift zonoid, weight, shift inequality, log-Sobolev inequality
Received by editor(s): September 4, 2013
Published electronically: January 26, 2015
Additional Notes: The second author was partially supported by the Leonard Euler program, DAAD project # 55518603
Article copyright: © Copyright 2015 American Mathematical Society

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