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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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Cite

BibTeXRIS
BibTeX
@misc{alias2026sobolev,
  title = {{Sobolev inequalities for sections and projections of log-concave functions}},
  author = {Luis J. Alías and Bernardo González Merino and Beatriz Marín Gimeno},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.32886},
  eprint = {2610.32886},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.32886},
}
RIS
TY  - UNPB
TI  - Sobolev inequalities for sections and projections of log-concave functions
AU  - Luis J. Alías
AU  - Bernardo González Merino
AU  - Beatriz Marín Gimeno
PY  - 2026
DA  - 2026/10/28
PB  - arXiv
DO  - 10.48550/arxiv.2610.32886
UR  - https://arxiv.org/abs/2610.32886
AB  - 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.
ER  -