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On sharp bounds for marginal densities of product measures

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Abstract

We discuss optimal constants in a recent result of Rudelson and Vershynin on marginal densities. We show that if f is a probability density on Rn of the form f(x) = П n i=1 f i (x i ), where each f i is a density on R, say bounded by one, then the density of any marginal π E (f) is bounded by 2k/2, where k is the dimension of E. The proof relies on an adaptation of Ball’s approach to cube slicing, carried out for functions. Motivated by inequalities for dual affine quermassintegrals, we also prove an isoperimetric inequality for certain averages of the marginals of such f for which the cube is the extremal case.

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Authors and Affiliations

  1. School of Mathematics, Georgia Institute of Technology, Atlanta, GA, 30319, USA

    Galyna Livshyts

  2. Department of Mathematics (Mailstop 3368), Texas A&M University, College Station, TX, 77843-3368, USA

    Grigoris Paouris

  3. Mathematics Department, University of Missouri, Columbia, MO, 65211, USA

    Peter Pivovarov

Authors
  1. Galyna Livshyts
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  2. Grigoris Paouris
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  3. Peter Pivovarov
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Correspondence to Galyna Livshyts.

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Livshyts, G., Paouris, G. & Pivovarov, P. On sharp bounds for marginal densities of product measures. Isr. J. Math. 216, 877–889 (2016). https://doi.org/10.1007/s11856-016-1431-5

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  • DOI: https://doi.org/10.1007/s11856-016-1431-5

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