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On a Linear Refinement of the Prékopa-Leindler Inequality
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- Journal:
- Canadian Journal of Mathematics / Volume 68 / Issue 4 / 01 August 2016
- Published online by Cambridge University Press:
- 20 November 2018, pp. 762-783
- Print publication:
- 01 August 2016
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- Article
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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.
