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High-Dimensional Convexity, Isoperimetry and Concentration via a Riemannian Vantage Point

Total Cost €


EC-Contrib. €






Project "CONC-VIA-RIEMANN" data sheet

The following table provides information about the project.


Organization address
city: HAIFA
postcode: 32000

contact info
title: n.a.
name: n.a.
surname: n.a.
function: n.a.
email: n.a.
telephone: n.a.
fax: n.a.

 Coordinator Country Israel [IL]
 Total cost 1˙194˙190 €
 EC max contribution 1˙194˙190 € (100%)
 Programme 1. H2020-EU.1.1. (EXCELLENT SCIENCE - European Research Council (ERC))
 Code Call ERC-2014-STG
 Funding Scheme ERC-STG
 Starting year 2015
 Duration (year-month-day) from 2015-10-01   to  2020-09-30


Take a look of project's partnership.

# participants  country  role  EC contrib. [€] 


 Project objective

'In recent years, the importance of superimposing the contribution of the measure to that of the metric, in determining the underlying space's (generalized Ricci) curvature, has been clarified in the works of Lott, Sturm, Villani and others, following the definition of Curvature-Dimension introduced by Bakry and Emery. We wish to systematically incorporate this important idea of considering the measure and metric in tandem, in the study of questions pertaining to isoperimetric and concentration properties of convex domains in high-dimensional Euclidean space, where a-priori there is only a trivial metric (Euclidean) and trivial measure (Lebesgue).

The first step of enriching the class of uniform measures on convex domains to that of non-negatively curved ('log-concave') measures in Euclidean space has been very successfully implemented in the last decades, leading to substantial progress in our understanding of volumetric properties of convex domains, mostly regarding concentration of linear functionals. However, the potential advantages of altering the Euclidean metric into a more general Riemannian one or exploiting related Riemannian structures have not been systematically explored. Our main paradigm is that in order to progress in non-linear questions pertaining to concentration in Euclidean space, it is imperative to cast and study these problems in the more general Riemannian context.

As witnessed by our own work over the last years, we expect that broadening the scope and incorporating tools from the Riemannian world will lead to significant progress in our understanding of the qualitative and quantitative structure of isoperimetric minimizers in the purely Euclidean setting. Such progress would have dramatic impact on long-standing fundamental conjectures regarding concentration of measure on high-dimensional convex domains, as well as other closely related fields such as Probability Theory, Learning Theory, Random Matrix Theory and Algorithmic Geometry.'


year authors and title journal last update
List of publications.
2016 Alexander V. Kolesnikov, Emanuel Milman
Riemannian metrics on convex sets with applications to Poincaré and log-Sobolev inequalities
published pages: , ISSN: 0944-2669, DOI: 10.1007/s00526-016-1018-3
Calculus of Variations and Partial Differential Equations 55/4 2019-05-27
2017 Nikos Dafnis, Grigoris Paouris
An inequality for moments of log-concave functions on Gaussian random vectors
published pages: 107-122, ISSN: , DOI: 10.1007/978-3-319-45282-1_7
Lecture Notes in Mathematics 2169 2019-05-27
2018 Emanuel Milman
Spectral estimates, contractions and hypercontractivity
published pages: 669-714, ISSN: 1664-039X, DOI:
Journal of Spectral Theory 8 (2) 2019-05-27
2017 Alexander V. Kolesnikov, Emanuel Milman
Sharp Poincar\'e-type inequality for the Gaussian measure on the boundary of convex sets
published pages: 221-234, ISSN: , DOI: 10.1007/978-3-319-45282-1_15
Lecture Notes in Mathematics 2169 2019-05-27
2017 Alexander V. Kolesnikov, Emanuel Milman
Brascamp–Lieb-Type Inequalities on Weighted Riemannian Manifolds with Boundary
published pages: 1680-1702, ISSN: 1050-6926, DOI: 10.1007/s12220-016-9736-5
The Journal of Geometric Analysis 27/2 2019-05-27

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