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Isoperimetric and Concentration Inequalities in High Dimensional Convex Spaces

"The proposed project lies at the intersection of the local theory of Banach spaces (more precisely ""Asymptotic Geometric Analysis""), Global Riemannian Geometry, and the study of isoperimetric and concentration properties of suc... ver más

31/05/2016

TECHNION ISRAEL I...

100K€

Presupuesto del proyecto: 100K€

Líder del proyecto

TECHNION ISRAEL INSTITUTE OF TECHNOLOGY No se ha especificado una descripción o un objeto social para esta compañía.

TRL 4-5

Fecha límite participación Sin fecha límite de participación.

Financiación concedida El organismo FP7 notifico la concesión del proyecto el día 2016-05-31 No tenemos la información de la convocatoria

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1 Participantes

TECHNION ISRAEL INSTITUTE O...

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Líder del proyecto

TECHNION ISRAEL INSTITUTE OF TECHNOLOGY No se ha especificado una descripción o un objeto social para esta compañía.

TRL 4-5

Presupuesto del proyecto 100K€

Fecha límite de participación Sin fecha límite de participación.

Descripción del proyecto "The proposed project lies at the intersection of the local theory of Banach spaces (more precisely ""Asymptotic Geometric Analysis""), Global Riemannian Geometry, and the study of isoperimetric and concentration properties of such spaces. We will study Riemannian manifolds endowed with a probability measure, whose (generalized Ricci) curvature is non-negative (""convex""), or more generally, bounded below (""semi-convex""); an important example is that of the uniform measure on a convex bounded domain in Euclidean space. Despite the immense diversity of these structures, even in the Euclidean case, it is known that they exhibit various unifying geometric and probabilistic properties. In this project, we are interested in various quantitative manifestations of the concentration of measure on these spaces, as their dimension tends to infinity. These include isoperimetric inequalities, providing lower bounds on the boundary measure of sets; Sobolev-type inequalities, such as the classical Poincar\'e (or Spectral-Gap) and logarithmic-Sobolev inequalities; and concentration of measure of various Lipschitz functionals, such as the distance functional. All present conjectures suggest that despite the great diversity, convexity and high-dimensionality serve as unifying forces which render all of these spaces not very different from some canonical ones, like the uniform measure on a Euclidean ball or hyper-cube. In recent years there has been much progress in the analysis of these and related questions. The proposed project intends to deepen and extend our qualitative and quantitative understanding of isoperimetric and concentration inequalities on high-dimensional convex and semi-convex manifolds-with-density in general, and on log-concave measures and convex bodies in particular."

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