Lift zonoid order and functional inequalities
Authors:
Alexei M. Kulik and Taras D. Tymoshkevych
Journal:
Theor. Probability and Math. Statist. 89 (2014), 83-99
MSC (2010):
Primary 26D10, 39B62, 47D07, 60E15, 60J60
DOI:
https://doi.org/10.1090/S0094-9000-2015-00937-6
Published electronically:
January 26, 2015
MathSciNet review:
3235177
Full-text PDF Free Access
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Additional Information
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.
References
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References
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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
Email:
kulik.alex.m@gmail.com
Taras D. Tymoshkevych
Affiliation:
National Taras Shevchenko University of Kyiv, Academician Glushkov Avenue 4-e, 03127, Kyiv, Ukraine
Email:
tymoshkevych@gmail.com
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