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Bounding marginal densities via affine isoperimetry
Author(s) -
Dann Susanna,
Paouris Grigoris,
Pivovarov Peter
Publication year - 2016
Publication title -
proceedings of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.899
H-Index - 65
eISSN - 1460-244X
pISSN - 0024-6115
DOI - 10.1112/plms/pdw026
Subject(s) - mathematics , bounding overwatch , affine transformation , bounded function , linear subspace , invariant (physics) , pure mathematics , ball (mathematics) , regular polygon , combinatorics , grassmannian , probability measure , discrete mathematics , mathematical analysis , geometry , artificial intelligence , computer science , mathematical physics
Let μ be a probability measure on R n with a bounded density f . We prove that the marginals of f on most subspaces are well‐bounded. For product measures, studied recently by Rudelson and Vershynin, our results show there is a trade‐off between the strength of such bounds and the probability with which they hold. Our proof rests on new affinely invariant extremal inequalities for certain averages of f on the Grassmannian and affine Grassmannian. These are motivated by Lutwak's dual affine quermassintegrals for convex sets. We show that key invariance properties of the latter, due to Grinberg, extend to families of functions. The inequalities we obtain can be viewed as functional analogues of results due to Busemann–Straus, Grinberg and Schneider. As an application, we show that without any additional assumptions on μ , any marginalπ E ( μ ) , or a small perturbation thereof, satisfies a nearly optimal small‐ball probability.