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.
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.