CELIPRO · Central Limit Properties of Convex Bodies
FP6 — Marie Curie Actions (Human Resources and Mobility)
- Duration
- 2005-12-01 → 2007-08-31
- EU contribution
- €152,548
- Participants
- 1
- Scheme
- EIF
Lines connect the coordinator with its partners.
Results in brief
Final Activity Report Summary - CELIPRO (Central Limit Properties of Convex Bodies)
The most natural shapes in geometry that one can draw and study are the convex bodies. If one is allowed to choose the coordinates one can assume that the convex bodies are isotropic, a classical notion that comes from mechanics. Experts in the field of convex geometry motivated by unsolved problems about isotropic bodies in the 90's started to speculate that isotropic convex bodies viewed as probability spaces, when the dimension grows large, share a property of regularity. In fact they behave almost identical. Various formulations of this principle can be titled 'central limit properties of high dimensional convex bodies'. The CELIPRO project was devoted to the effort to understand this problem and to reveal this principle. Indeed significant progress was made and several formulations of the question were proved. Here we choose to present the one that is known that the estimate established is optimal: Let K be an isotropic convex body in n dimensions and let I be the mean radius of the volume. Then, all the volume except an insignificant part ( as small as exponential to minus square root of the dimension) lies inside a spherical cell around the mean I. Isotropic convex bodies in high dimension have an extreme diversity if viewed in geometric terms like diameter, facets, curvature. Although the result shows that as probability spaces are almost indistinguishable: If one will try to recognise it or reconstruct it by choosing random points from it (even if he will take exponential to the square root of the dimension many points) he will be unable to distinguish it from the Euclidean ball. This phenomenon appears to be an inherent component of high dimensional geometry. The result found already applications in the field and in other branches of mathematics studying high dimensional systems.
Data: CORDIS, © European Union
Project objective
Asymptotic convex geometry is concerned with the geometric and linear properties of finine dimensional normed spaces or convex bodies, the emphasis being on the asymptotic behavior of various quantitative parameters as the dimension grows to infinity. It i s a central component of an emerging interdisciplinary area, which deals with high dimensional phenomena and lies on the intersection of geometry, analysis, probability and combinatorics. An intriguing question in asymptotic convex geometry is to understan d the central limit properties of convex bodies. The problem, which is increasingly attracting the attention and the efforts of many researchers, can be vaguely formulated as follows : is it true that every convex body of high dimension has most of its mar ginal distributions essentially Gaussian ? The problem is naturally linked to the classical slicing problem and to the study of volume concentration on isotropic convex bodies. Pajor has done pioneering work on isotropicity and he is one of the leading exp erts in asymptotic convex geometry. Paouris has devoted a lot of effort on this particular subject : he has already contributed to the picture and brings interesting new ideas and questions. The area is open and promising, and one should expect important n ew steps. A second main goal of this project is to help the participant researcher to learn new techniques in his field, but also to have exposure to a variety of influences and to identify new fields in which he might work in the future. The Department of Mathematics at the University of Marne-la-Vallee contains a leading team on convex geometric analysis but also on deviation estimates in probability, optimal transortation, entropy growth and their relation to geometry. This makes it an ideal place of t he training needs of Paouris. This will hopefully secure his academic future but will also have a positive influence on the research community in Greece.
Original text from CORDIS.
Participants
- UNIVERSITE DE MARNE LA VALLEE · MARNE LA VALLEECoordinatorFrance
Links
Data: CORDIS, © European Union