31 research outputs found
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Mixed volumes and mixed integrals
In recent years, mathematicians have developed new approaches to study convex sets: instead of considering convex sets themselves, they explore certain functions or measures that are related to them. Problems from convex geometry become thereby accessible to analytic and probabilistic tools, and we can use these tools to make progress on very difficult open problems. We discuss in this Snapshot such a functional extension of some âvolumesâ which measure how âbigâ a set is. We recall the construction of âintrinsic volumesâ, discuss the fundamental inequalities between them, and explain the functional extensions of these results
New Brunn--Minkowski and functional inequalities via convexity of entropy
We study the connection between the concavity properties of a measure
and the convexity properties of the associated relative entropy along optimal transport. As a corollary we prove a new dimensional
Brunn-Minkowski inequality for centered star-shaped bodies, when the measure
is log-concave with a p-homogeneous potential (such as the Gaussian
measure). Our method allows us to go beyond the usual convexity assumption on
the sets that is fundamentally essential for the standard
differential-geometric technique in this area.
We then take a finer look at the convexity properties of the Gaussian
relative entropy, which yields new functional inequalities. First we obtain
curvature and dimensional reinforcements to Otto--Villani's "HWI" inequality in
the Gauss space, when restricted to even strongly log-concave measures. As
corollaries, we obtain improved versions of Gross' logarithmic Sobolev
inequality and Talgrand's transportation cost inequality in this setting.Comment: 29 pages; Added some historical references (Remark 3), comments on
the sharpness of our log-Sobolev and Talagrand inequalities (in Examples 5.6,
5.9), and technical details. Comments welcome
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
Stability and the equality case in the B-theorem
In this paper, we show the stability, and characterize the equality cases in
the strong B-inequality of Cordero-Erasquin, Fradelizi and Maurey
\cite{B-conj}. As an application, we establish uniqueness of Bobkov's maximal
Gaussian measure position from \cite{Bobkov-Mpos}