335 research outputs found
On matrix-valued log-concavity and related Prekopa and Brascamp-Lieb inequalities
We propose a new, self-contained, approach to H. Raufi's extension of
Prekopa's theorem for matrix-valued log-concave functions. Along the way, new
related inequalities are established, in particular a Brascamp-Lieb variance
inequality for matrix weights
Invariances in variance estimates
We provide variants and improvements of the Brascamp-Lieb variance inequality
which take into account the invariance properties of the underlying measure.
This is applied to spectral gap estimates for log-concave measures with many
symmetries and to non-interacting conservative spin systems
The (B) conjecture for uniform measures in the plane
We prove that for any two centrally-symmetric convex shapes , the function is log-concave. This
extends a result of Cordero-Erausquin, Fradelizi and Maurey in the two
dimensional case. Possible relaxations of the condition of symmetry are
discussed.Comment: 10 page
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
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