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On a Linear Refinement of the Prékopa-Leindler Inequality

Published online by Cambridge University Press:  20 November 2018

Andrea Colesanti ,
Eugenia Saorín Gómez  and
Jesús Yepes Nicolás
Andrea Colesanti
Affiliation:
Dipartimento di Matematica “U. Dini”, Viale Morgagni 67/A, 50134-Firenze, Italy e-mail: colesant@math.unifi.it
Eugenia Saorín Gómez
Affiliation:
Institut für Algebra und Geometrie, Otto-von-Guericke Universität Magdeburg, Universitätsplatz 2, D-39106-Magdeburg, Germany e-mail: eugenia.saorin@ovgu.de
Jesús Yepes Nicolás
Affiliation:
Departamento de Matemáticas, Universidad de Murcia, Campus de Espinardo, 30100-Murcia, Spain e-mail: jesus.yepes@um.es
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Abstract

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If $f,\,g:\,{{\mathbb{R}}^{n}}\,\to \,{{\mathbb{R}}_{\ge 0}}$ are non-negative measurable functions, then the Prékopa–Leindler inequality asserts that the integral of the Asplund sum (provided that it is measurable) is greater than or equal to the 0-mean of the integrals of $f$ and $g$. In this paper we prove that under the sole assumption that $f$ and $g$ have a common projection onto a hyperplane, the Prékopa–Leindler inequality admits a linear refinement. Moreover, the same inequality can be obtained when assuming that both projections (not necessarily equal as functions) have the same integral. An analogous approach may be also carried out for the so-called Borell-Brascamp-Lieb inequality.

Keywords

52A4026D1526B25Prékopa–Leindler inequalitylinearityAsplund sumprojectionsBorell–Brascamp–Lieb inequality
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 68 , Issue 4 , 01 August 2016 , pp. 762 - 783
Copyright
Copyright © Canadian Mathematical Society 2016

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