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Rogers–Shephard and local Loomis–Whitney type inequalities

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Abstract

We provide functional analogues of the classical geometric inequality of Rogers and Shephard on products of volumes of sections and projections. As a consequence we recover (and obtain some new) functional versions of Rogers–Shephard type inequalities as well as some generalizations of the geometric Rogers–Shephard inequality in the case where the subspaces intersect. These generalizations can be regarded as sharp local reverse Loomis–Whitney inequalities. We also obtain a sharp local Loomis–Whitney inequality.

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Acknowledgements

Part of this work was carried out at the IMUS of the University of Sevilla and the authors are grateful for the hospitality and the support provided by the IMUS and MINECO Grant MTM2015-63699-P during their stay.

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Correspondence to Carlos Hugo Jiménez.

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Communicated by Andreas Thom.

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David Alonso-Gutiérrez is partially supported by MINECO project MTM2016-77710-P, by DGA E26_17R, and by IUMA. Shiri Artstein-Avidan is partially supported by ISF Grant number 665/15. This research is a result of the activity developed within the framework of the Programme in Support of Excellence Groups of the Región de Murcia, Spain, by Fundación Séneca, Science and Technology Agency of the Región de Murcia. Bernardo González Merino was partially supported by Fundación Séneca project 19901/GERM/15, Spain. C. Hugo Jiménez is supported by CNPq and the program Incentivo à produtividade em ensino e pesquisa from the PUC-Rio. Bernardo González Merino and Rafael Villa was partially supported by MINECO project reference MTM2015-63699-P, Spain.

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Alonso-Gutiérrez, D., Artstein-Avidan, S., González Merino, B. et al. Rogers–Shephard and local Loomis–Whitney type inequalities. Math. Ann. 374, 1719–1771 (2019). https://doi.org/10.1007/s00208-019-01834-3

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  • DOI: https://doi.org/10.1007/s00208-019-01834-3

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