Simon Zugmeyer
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Transport-entropy and functional forms of Blaschke–Santaló inequalities Open
We explore alternative functional or transport-entropy formulations of the Blaschke–Santaló inequality. In particular, we obtain new Blaschke–Santaló inequalities for s -concave functions. We also obtain new sharp symmetrized transport-ent…
View article: Some obstructions to contraction theorems on the half-sphere
Some obstructions to contraction theorems on the half-sphere Open
Caffarelli's contraction theorem states that probability measures with uniformly logconcave densities on R d can be realized as the image of a standard Gaussian measure by a globally Lipschitz transport map. We discuss some counterexamples…
Transport-entropy forms of direct and Converseblaschke-Santal{ó} inequalities Open
We explore alternative functional or transport-entropy formulations of the Blaschke-Santal{ó} inequality and of its conjectured counterpart due to Mahler. In particular, we obtain new direct and reverse Blaschke-Santal{ó} inequalities for …
Sobolev’s inequality under a curvature-dimension condition Open
In this note we present a new proof of Sobolev’s inequality under a uniform lower bound of the Ricci curvature. This result was initially obtained in 1983 by Ilias. Our goal is to present a very short proof, to give a review of the famous …
A conformal geometric point of view on the Caffarelli-Kohn-Nirenberg inequality Open
We are interested in the Caffarelli-Kohn-Nirenberg inequality (CKN in short), introduced by these authors in 1984. We explain why the CKN inequality can be viewed as a Sobolev inequality on a weighted Riemannian manifold. More precisely, w…
Transport proofs of some functional inverse Santal{ó} inequalities Open
In this paper, we present a simple proof of a recent result of the second author which establishes that functional inverse-Santal{ó} inequalities follow from Entropy-Transport inequalities. Then, using transport arguments together with ele…
A family of Beckner inequalities under various curvature-dimension conditions Open
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Sobolev's inequality under a curvature-dimension condition Open
In this note we present a new proof of Sobolev's inequality under a uniform lower bound of the Ricci curvature. This result was initially obtained in 1983 by Ilias. Our goal is to present a very short proof, to give a review of the famous …
Entropy flows and functional inequalities in convex sets Open
We revisit entropy methods to prove new sharp trace logarithmic Sobolev and sharp Gagliardo-Nirenberg-Sobolev inequalities on the half space, with a focus on the entropy inequality itself and not the actual flow, allowing for somewhat robu…
A family of Beckner inequalities under various curvature-dimension\n conditions Open
In this paper, we offer a proof for a family of functional inequalities\ninterpolating between the Poincar{\\'e} and the logarithmic Sobolev (standard\nand weighted) inequalities. The proofs rely both on entropy flows and on a\nCD($\\rho$,…
Sharp trace Gagliardo–Nirenberg–Sobolev inequalities for convex cones, and convex domains Open
We find a new sharp trace Gagliardo–Nirenberg–Sobolev inequality on convex cones, as well as a sharp weighted trace Sobolev inequality on epigraphs of convex functions. This is done by using a generalized Borell–Brascamp–Lieb inequality, c…
Sharp trace Gagliardo-Nirenberg-Sobolev inequalities for convex cones, and convex domains Open
We find a new sharp trace Gagliardo-Nirenberg-Sobolev inequality on convex cones, aswell as a weighted sharp trace Sobolev inequality on epigraphs of convex functions. This is done by using a generalized Borell-Brascamp-Lieb inequality, co…