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# On a Linear Refinement of the Prékopa-Leindler Inequality

Published:2016-04-21
Printed: Aug 2016
• Andrea Colesanti,
Dipartimento di Matematica "U. Dini", Viale Morgagni 67/A, 50134-Firenze, Italy
• Eugenia Saorín Gómez,
Institut für Algebra und Geometrie, Otto-von-Guericke Universität Magdeburg, Universitätsplatz 2, D-39106-Magdeburg, Germany
• Jesus Yepes Nicolás,
Departamento de Matemáticas, Universidad de Murcia, Campus de Espinardo, 30100-Murcia, Spain
 Format: LaTeX MathJax PDF

## Abstract

If $f,g:\mathbb{R}^n\longrightarrow\mathbb{R}_{\geq0}$ 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 or equal than 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: Prékopa-Leindler inequality, linearity, Asplund sum, projections, Borell-Brascamp-Lieb inequality
 MSC Classifications: 52A40 - Inequalities and extremum problems 26D15 - Inequalities for sums, series and integrals 26B25 - Convexity, generalizations

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