Abstract
A sharp Poincaré-type inequality is derived for the restriction of the Gaussian measure on the boundary of a convex set. In particular, it implies a Gaussian mean-curvature inequality and a Gaussian iso-second-variation inequality. The new inequality is nothing but an infinitesimal equivalent form of Ehrhard’s inequality for the Gaussian measure. While Ehrhard’s inequality does not extend to general CD(1, ∞) measures, we formulate a sufficient condition for the validity of Ehrhard-type inequalities for general measures on \(\mathbb{R}^{n}\) via a certain property of an associated Neumann-to-Dirichlet operator.
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Acknowledgements
The first named author was supported by the RFBR project 17-01-00662, and the DFG project RO 1195/12-1. The second named author is supported by BSF (grant no. 2010288) and Marie-Curie Actions (grant no. PCIG10-GA-2011-304066). The article was prepared within the framework of the Academic Fund Program at the National Research University Higher School of Economics (HSE) in 2017–2018 (grant no. 17-01-0102) and by the Russian Academic Excellence Project “5-100”. The research leading to these results is part of a project that has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement no. 637851).
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Kolesnikov, A.V., Milman, E. (2017). Sharp Poincaré-Type Inequality for the Gaussian Measure on the Boundary of Convex Sets. In: Klartag, B., Milman, E. (eds) Geometric Aspects of Functional Analysis. Lecture Notes in Mathematics, vol 2169. Springer, Cham. https://doi.org/10.1007/978-3-319-45282-1_15
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DOI: https://doi.org/10.1007/978-3-319-45282-1_15
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