Final Report Summary - ICIHDCS (Isoperimetric and Concentration Inequalities in High-Dimensional Convex Spaces)
The goal of the Marie-Curie CIG project "Isoperimetric and Concentration Inequalities in High-Dimensional Convex Spaces (ICIHDCS)" I have set out to achieve was to enhance our qualitative and quantitative understanding of isoperimetric inequalities and isoperimetric minimizers on high-dimensional convex and semi-convex manifolds-with-density in general, and on log-concave measures and convex bodies in Euclidean space in particular. The emphasis was on obtaining the right dependence on the dimension in our sought-after inequalities (or identifying "almost" minimizers up to dimension-independent constants), and on applications of such advancements. We emphasize that isoperimetry is only one particular facet in the study of interaction between measure and metric, and that other related ways exist to measure this interplay, such as by means of Sobolev-type inequalities (Poincare, log-Sobolev, etc..) and concentration inequalities (measuring the large deviation distribution of Lipschitz functionals on our space). My goal was to progress in our understanding of any of these other related tiers as well.
Our motivation comes from several long-standing open problems regarding convex bodies in Euclidean space: the isoperimetric Kannan-Lovasz-Simonovits conjecture on the Cheeger constant (the latter is well-known to have an equivalent spectral formulation involving the Poincare inequality and the spectral-gap); the thin-shell conjecture on concentration of the Euclidean norm about its expected value; and Bourgain's Slicing problem on the relation between volume and the variance of linear functionals on convex domains, as measured by the Slicing (or isotropic) constant.
These types of questions are very challenging because high dimensional space tends to contradict our intuition: for instance, most of the mass of a high-dimensional ball lies almost entirely near the ball's boundary, and not near its center. The situation with more general convex bodies is far less understood, and quantitatively determining where most of their mass lies is important for various applications: creating algorithms to tell convex bodies apart from a ball, understanding the distribution of a point randomly selected inside the body, deciding how to most efficiently cut the convex body in half, and so on. Our project addresses some of the fundamental questions on the distribution of mass inside a general high-dimensional convex body and the interrelations between the various questions themselves. For instance, can we cover most of a (non-degenerate) convex body by a ball of essentially the same volume? can we cover it by a very narrow annulus? can we bisect it into two halves efficiently just by using a single blow of a knife? These questions lie at the very heart of the theory of Asymptotic Geometric Analysis, and serve as tools in other fields such as Probability Theory, Statistical Physics, Random Matrix Theory, Learning Theory, Algorithmic Geometry, Combinatorics and Complexity.
During the project, we have indeed derived applications of the isoperimetric-concentration correspondence I had previously discovered to problems in Statistical Mechanics. We continued our study of the Slicing Problem, and derived new connections between the isotropic constant and other natural parameters associated to convex bodies, such as the mean-width and the mean-norm, establishing in some cases essentially best-possible conjectured relations. We continued to emphasize the importance of extending the known results and tools from the Euclidean to the Riemannian setting, and succeeded to apply the L^2 method to obtain Poincare-type inequalities on the boundary of convex domains of weighted Riemannian manifolds; moreover, the resulting Brunn-Minkowski type inequality enabled us to construct a novel geometric-flow in the Riemannian setting, which may be interpreted as the Riemannian analogue of the Minkowski addition, a notion previously confined to the linear setting. The resulting Poincare-type inequalities were new even in the classical case of the Gaussian measures in Euclidean space. We have also shown that the Riemannian setting is useful for obtaining Poincare-type inequalities for measures in the purely Euclidean setting, by deforming the Euclidean metric to a Riemannian one, suitably chosen to be compatible with the measure in question. Other closely related methods such as the Lyapunov method enabled us to deduce various spectral-gap estimates on convex domains in Euclidean space.
We have also managed to extend the notions of generalized curvature and dimension to include negative (!) dimension, and showed that various natural weighted manifolds have negative generalized dimension. We extended many of the known spectral and concentration results from the positive dimension case to the negative one, and in both cases derived novel sharp (i.e. best-possible) isoperimetric inequalities, for weighted manifolds having a lower bound on their generalized curvature, and upper bounds on their generalized dimension and diameter. We have also formulated and investigated a conjecture stating that such weighted manifolds are always the image of a contracting map from a corresponding model space, explaining many of the isoperimetric, spectral and volume-comparison properties these spaces have in common with the model space.
The above results have been described in 16 published, accepted and submitted papers, 9 of which were written jointly with my collaborators Franck Barthe, Apostolos Giannopoulos, Alexander Kolesnikov and Liran Rotem.
During the project's span, I have given invited talks in over 20 international conferences, as well as numerous additional colloquia and seminars. I co-edited (together with Bo'az Klartag) 2 volumes of Springer's Lecture Notes in Mathematics series entitled "Geometric and Functional Analysis, Israel Seminar", and have joined the editorial board of Journal of Functional Analysis. I taught 4 advanced graduate-level courses at the Technion and one mini-course at the Hausdorff Institute of Mathematics (Bonn) in my field of expertise, and have supervised two post-doctoral fellows and one graduate student.
I have been promoted to the rank of tenured Associate Professor, awarded the Cooper Award for Academic Excellence by the Technion, and the 2016 Anna and Lajos Erdos Prize in Mathematics by the Israeli Mathematical Union.
Our motivation comes from several long-standing open problems regarding convex bodies in Euclidean space: the isoperimetric Kannan-Lovasz-Simonovits conjecture on the Cheeger constant (the latter is well-known to have an equivalent spectral formulation involving the Poincare inequality and the spectral-gap); the thin-shell conjecture on concentration of the Euclidean norm about its expected value; and Bourgain's Slicing problem on the relation between volume and the variance of linear functionals on convex domains, as measured by the Slicing (or isotropic) constant.
These types of questions are very challenging because high dimensional space tends to contradict our intuition: for instance, most of the mass of a high-dimensional ball lies almost entirely near the ball's boundary, and not near its center. The situation with more general convex bodies is far less understood, and quantitatively determining where most of their mass lies is important for various applications: creating algorithms to tell convex bodies apart from a ball, understanding the distribution of a point randomly selected inside the body, deciding how to most efficiently cut the convex body in half, and so on. Our project addresses some of the fundamental questions on the distribution of mass inside a general high-dimensional convex body and the interrelations between the various questions themselves. For instance, can we cover most of a (non-degenerate) convex body by a ball of essentially the same volume? can we cover it by a very narrow annulus? can we bisect it into two halves efficiently just by using a single blow of a knife? These questions lie at the very heart of the theory of Asymptotic Geometric Analysis, and serve as tools in other fields such as Probability Theory, Statistical Physics, Random Matrix Theory, Learning Theory, Algorithmic Geometry, Combinatorics and Complexity.
During the project, we have indeed derived applications of the isoperimetric-concentration correspondence I had previously discovered to problems in Statistical Mechanics. We continued our study of the Slicing Problem, and derived new connections between the isotropic constant and other natural parameters associated to convex bodies, such as the mean-width and the mean-norm, establishing in some cases essentially best-possible conjectured relations. We continued to emphasize the importance of extending the known results and tools from the Euclidean to the Riemannian setting, and succeeded to apply the L^2 method to obtain Poincare-type inequalities on the boundary of convex domains of weighted Riemannian manifolds; moreover, the resulting Brunn-Minkowski type inequality enabled us to construct a novel geometric-flow in the Riemannian setting, which may be interpreted as the Riemannian analogue of the Minkowski addition, a notion previously confined to the linear setting. The resulting Poincare-type inequalities were new even in the classical case of the Gaussian measures in Euclidean space. We have also shown that the Riemannian setting is useful for obtaining Poincare-type inequalities for measures in the purely Euclidean setting, by deforming the Euclidean metric to a Riemannian one, suitably chosen to be compatible with the measure in question. Other closely related methods such as the Lyapunov method enabled us to deduce various spectral-gap estimates on convex domains in Euclidean space.
We have also managed to extend the notions of generalized curvature and dimension to include negative (!) dimension, and showed that various natural weighted manifolds have negative generalized dimension. We extended many of the known spectral and concentration results from the positive dimension case to the negative one, and in both cases derived novel sharp (i.e. best-possible) isoperimetric inequalities, for weighted manifolds having a lower bound on their generalized curvature, and upper bounds on their generalized dimension and diameter. We have also formulated and investigated a conjecture stating that such weighted manifolds are always the image of a contracting map from a corresponding model space, explaining many of the isoperimetric, spectral and volume-comparison properties these spaces have in common with the model space.
The above results have been described in 16 published, accepted and submitted papers, 9 of which were written jointly with my collaborators Franck Barthe, Apostolos Giannopoulos, Alexander Kolesnikov and Liran Rotem.
During the project's span, I have given invited talks in over 20 international conferences, as well as numerous additional colloquia and seminars. I co-edited (together with Bo'az Klartag) 2 volumes of Springer's Lecture Notes in Mathematics series entitled "Geometric and Functional Analysis, Israel Seminar", and have joined the editorial board of Journal of Functional Analysis. I taught 4 advanced graduate-level courses at the Technion and one mini-course at the Hausdorff Institute of Mathematics (Bonn) in my field of expertise, and have supervised two post-doctoral fellows and one graduate student.
I have been promoted to the rank of tenured Associate Professor, awarded the Cooper Award for Academic Excellence by the Technion, and the 2016 Anna and Lajos Erdos Prize in Mathematics by the Israeli Mathematical Union.