41 research outputs found
Improved log-concavity for rotationally invariant measures of symmetric convex sets
We prove that the (B) conjecture and the Gardner-Zvavitch conjecture are true
for all log-concave measures that are rotationally invariant, extending
previous results known for Gaussian measures. Actually, our result apply beyond
the case of log-concave measures, for instance to Cauchy measures as well. For
the proof, new spectral inequalities are obtained for even probability measures
that are log-concave with respect to a rotationally invariant measure.Comment: typos and references fixe
Plurisubharmonic geodesics and interpolating sets
We apply a notion of geodesics of plurisubharmonic functions to interpolation
of compact subsets of . Namely, two non-pluripolar, polynomially closed,
compact subsets of are interpolated as level sets
for the geodesic between their relative extremal functions with respect
to any ambient bounded domain. The sets are described in terms of certain
holomorphic hulls. In the toric case, it is shown that the relative
Monge-Amp\`ere capacities of satisfy a dual Brunn-Minkowski inequality.Comment: Minor changes. Final version, to appear in Arch. Mat
Volume of the polar of random sets and shadow systems
We obtain optimal inequalities for the volume of the polar of random sets,
generated for instance by the convex hull of independent random vectors in
Euclidean space. Extremizers are given by random vectors uniformly distributed
in Euclidean balls. This provides a random extension of the Blaschke-Santalo
inequality which, in turn, can be derived by the law of large numbers. The
method involves generalized shadow systems, their connection to Busemann type
inequalities, and how they interact with functional rearrangement inequalities
New sharp Gagliardo-Nirenberg-Sobolev inequalities and an improved Borell-Brascamp-Lieb inequality
We propose a new Borell-Brascamp-Lieb inequality which leads to novel sharp
Euclidean inequalities such as Gagliardo-Nirenberg-Sobolev inequalities in R^n
and in the half-space R^n\_+. This gives a new bridge between the geometric
pont of view of the Brunn-Minkowski inequality and the functional point of view
of the Sobolev type inequalities. In this way we unify, simplify and results by
S. Bobkov-M. Ledoux, M. del Pino-J. Dolbeault and B. Nazaret
Asymmetric covariance estimates of Brascamp-Lieb type and related inequalities for log-concave measures
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