## Periodic Reporting for period 1 - CONC-VIA-RIEMANN (High-Dimensional Convexity, Isoperimetry and Concentration via a Riemannian Vantage Point)

**Reporting period:**2015-10-01

**to**2017-03-31

## Summary of the context and overall objectives of the project

The main aspect of our project is to emphasize how the vantage point of Riemannian Geometry can be used to enhance our qualitative and quantitative understanding of the concentration of volume and the isoperimetric properties on log-concave measures and convex bodies in Euclidean space. Our motivation comes from several long-standing open problems in these directions: the isoperimetric Kannan--Lov\'asz--Simonovits conjecture on the Cheeger constant (the latter is well-known to have an equivalent spectral formulation involving the Poincar\'e 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.

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 the other closely related fields mentioned above.

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.

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 the other closely related fields mentioned above.

## Work performed from the beginning of the project to the end of the period covered by the report and main results achieved so far

"Together with A. Kolesnikov, we have found a unified framework for deducing numerous Poincar\'e-type inequalities, which are known to be intimately connected to the Brunn--Minkowski inequality, a fundamental inequality in Convexity Theory which lies at the heart of our project. This framework works equally well on a Riemannian manifold equipped with a density (``weighted-manifold"") and having a boundary, and under various combinations of boundary conditions of the domain (convex, mean-convex) and the function (Neumann, Dirichlet), we obtain new Poincar\'e-type inequalities on the manifold and on its boundary. For instance, we may handle Neumann conditions on a mean-convex domain, and obtain generalizations to the manifold-with-density setting of a purely Euclidean inequality of Colesanti, which yields a Brunn--Minkowski concavity result for convex domains in the manifold setting. All previously known inequalities due to e.g. Lichnerowicz, Brascamp--Lieb and Veysseire are also recovered in a single, unified framework. A particularly intriguing aspect of this work was the discovery of a new geometric evolution equation, which seems to be the Riemannian analogue of Minkowski addition, a notion previously confined to the linear setting.

Armed with these new tools in the Riemannian setting, we proceeded with A. Kolesnikov to obtain new concentration results in the purely Euclidean setting. Specifically, we were interested in obtaining Poincar\'e and log-Sobolev type inequalities for a given a probability measure $\mu$ supported on a convex subset of Euclidean space. Our idea was to change the Euclidean metric to a more general Riemannian one, adapted in a certain sense to $\mu$, and perform our analysis on the resulting weighted Riemannian manifold. The types of metrics we considered are Hessian metrics (intimately related to associated optimal-transport problems), product metrics (which are very useful when m is unconditional, i.e. invariant under reflections with respect to the coordinate hyperplanes), and metrics conformal to the Euclidean one, which have not been previously explored in this context. Invoking tools such as Riemannian generalizations of the Brascamp--Lieb inequality and the Bakry--\'Emery criterion, and passing back to the original Euclidean metric, we obtained various weighted inequalities for $\mu$: refined and entropic versions of the Brascamp--Lieb inequality, weighted Poincar\'e and log-Sobolev inequalities, Hardy-type inequalities, etc... In particular, we recovered the various known estimates on the thin-shell and spectral-gap (or Cheeger constant) of unconditional log-concave measures obtained by Klartag, and obtained new improved ones under some further assumptions. Key to our analysis is the positivity of the associated Lichnerowicz--Bakry--\'Emery generalized Ricci curvature tensor, and the convexity of the auxiliary weighted-manifold. In some cases, we could only ensure that the latter manifold is (generalized) mean-convex, resulting in additional boundary terms in our inequalities.

A well-known tool to transfer isoperimetric, functional and concentration inequalities from a source model metric-measure space to a target one, is to find a contraction pushing forward the measure on the former onto the one on the latter. A classical example is given by Caffarelli's Contraction Theorem, which enables transferring all the good properties of a source Gaussian measure to a target measure which is more log-concave than the source one. Consequently, it is of prime importance to find interesting contractions. In our work pertaining to the interplay between contractions and lower bounds on the Ricci curvature, we conjectured that given an $n$-dimensional compact Riemannian manifold with Ricci curvature bounded below by $K > 0$, there should exist a contracting map from the $n$-sphere having Ricci curvature equal to $K$ to that manifold, which pushes forward the normalized volume measure of the former onto"

Armed with these new tools in the Riemannian setting, we proceeded with A. Kolesnikov to obtain new concentration results in the purely Euclidean setting. Specifically, we were interested in obtaining Poincar\'e and log-Sobolev type inequalities for a given a probability measure $\mu$ supported on a convex subset of Euclidean space. Our idea was to change the Euclidean metric to a more general Riemannian one, adapted in a certain sense to $\mu$, and perform our analysis on the resulting weighted Riemannian manifold. The types of metrics we considered are Hessian metrics (intimately related to associated optimal-transport problems), product metrics (which are very useful when m is unconditional, i.e. invariant under reflections with respect to the coordinate hyperplanes), and metrics conformal to the Euclidean one, which have not been previously explored in this context. Invoking tools such as Riemannian generalizations of the Brascamp--Lieb inequality and the Bakry--\'Emery criterion, and passing back to the original Euclidean metric, we obtained various weighted inequalities for $\mu$: refined and entropic versions of the Brascamp--Lieb inequality, weighted Poincar\'e and log-Sobolev inequalities, Hardy-type inequalities, etc... In particular, we recovered the various known estimates on the thin-shell and spectral-gap (or Cheeger constant) of unconditional log-concave measures obtained by Klartag, and obtained new improved ones under some further assumptions. Key to our analysis is the positivity of the associated Lichnerowicz--Bakry--\'Emery generalized Ricci curvature tensor, and the convexity of the auxiliary weighted-manifold. In some cases, we could only ensure that the latter manifold is (generalized) mean-convex, resulting in additional boundary terms in our inequalities.

A well-known tool to transfer isoperimetric, functional and concentration inequalities from a source model metric-measure space to a target one, is to find a contraction pushing forward the measure on the former onto the one on the latter. A classical example is given by Caffarelli's Contraction Theorem, which enables transferring all the good properties of a source Gaussian measure to a target measure which is more log-concave than the source one. Consequently, it is of prime importance to find interesting contractions. In our work pertaining to the interplay between contractions and lower bounds on the Ricci curvature, we conjectured that given an $n$-dimensional compact Riemannian manifold with Ricci curvature bounded below by $K > 0$, there should exist a contracting map from the $n$-sphere having Ricci curvature equal to $K$ to that manifold, which pushes forward the normalized volume measure of the former onto"

## Progress beyond the state of the art and expected potential impact (including the socio-economic impact and the wider societal implications of the project so far)

All of the above mentioned works constitute original research which go well beyond the previously known state of the art.