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arXiv · 2610.32886

Sobolev inequalities for sections and projections of log-concave functions

Luis J. Alías·Bernardo González Merino·Beatriz Marín Gimeno
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Abstract

Higher order sharp Sobolev and local Sobolev type inequalities for log-concave functions are established. The new inequalities directly imply the isoperimetric inequalities by Bollobás and Thomason in 1995, by Liakopoulos in 2019, and the recently discovered local isoperimetric inequalities from [1]. In order to do so, we make use of three ingredients. First, sets of functional inequalities proven by Alonso-Gutiérrez, Bernués, Brazitikos and Carbery in 2021. Second, a multidimensional polynomial optimization result that originated in [1] and was initiated by Bollobás and Thomason. Third, a new integral formula for the projection of a log-concave function that links the geometric part with the more functional versions via some dependence on the Hessian matrix restricted to subspaces.

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Luis J. Alías, Bernardo González Merino, Beatriz Marín Gimeno. 2026-10-23. Sobolev inequalities for sections and projections of log-concave functions. https://arxiv.org/abs/2610.32886

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