A symmetrization inequality shorn of symmetry
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- by Michael Christ and Dominique Maldague PDF
- Trans. Amer. Math. Soc. 373 (2020), 5997-6028 Request permission
Abstract:
An inequality of Brascamp-Lieb-Luttinger and of Rogers states that among subsets of Euclidean space $\mathbb {R}^d$ of specified Lebesgue measures, (tuples of) balls centered at the origin are maximizers of certain functionals defined by multidimensional integrals. For $d>1$, this inequality only applies to functionals invariant under a diagonal action of $\operatorname {Sl}(d)$. We investigate functionals of this type, and their maximizers, in perhaps the simplest situation in which $\operatorname {Sl}(d)$ invariance does not hold. Assuming a more limited symmetry encompassing dilations but not rotations, we show under natural hypotheses that maximizers exist, and, moreover, that there exist distinguished maximizers whose structure reflects this limited symmetry. For small perturbations of the $\operatorname {Sl}(d)$–invariant framework we show that these distinguished maximizers are strongly convex sets with infinitely differentiable boundaries. It is shown that in the absence of partial symmetry, maximizers fail to exist for certain arbitrarily small perturbations of $\operatorname {Sl}(d)$–invariant structures.References
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Additional Information
- Michael Christ
- Affiliation: Department of Mathematics, University of California, Berkeley, California 94720-3840
- MR Author ID: 48950
- Email: mchrist@berkeley.edu
- Dominique Maldague
- Affiliation: Department of Mathematics, University of California, Berkeley, California 94720-3840
- MR Author ID: 1316122
- Email: dmal@math.berkeley.edu
- Received by editor(s): January 4, 2019
- Received by editor(s) in revised form: February 25, 2020
- Published electronically: May 26, 2020
- Additional Notes: The first author was supported in part by NSF grant DMS-1363324.
The second author was supported by an NSF graduate research fellowship under Grant No. DGE 1106400. - © Copyright 2020 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 373 (2020), 5997-6028
- MSC (2010): Primary 26D15; Secondary 42B99
- DOI: https://doi.org/10.1090/tran/8145
- MathSciNet review: 4127899