arXiv research

Kim-Milman flow map stable under regular target measures

Nearly spherical, positively curved surfaces are mapped from a sphere.

Researchers found counterexamples to conjectures about optimal transport maps on curved spaces.

Maps from spheres and disks to convex shapes via curvature flow.

The paper proves waist inequalities for convex bodies and their linear images.

The present note is a result of an on-going investigation into the logarithmic Brunn-Minkowski inequality. We obtain lower estimates on the volume product for convex bodies in $\mathbb{R}^n$ not necessarily symmetric with respect to the origin from a modified logarithmic Brunn-Minkowski inequality.

Geometric proof of Lojasiewicz inequalities for $C^1$ functions with simple normal crossings.

In this paper, we study the Lévy-Milman concentration phenomenon of 1-Lipschitz maps into infinite dimensional metric spaces. Our main theorem asserts that the concentration to an infinite dimensional $\ell^p$-ball with the $\ell^q$-distance function for $1\leq p<q\leq +\infty$ is equivalent to the concentration to the…

In this article a class of closed convex sets in the Euclidean $n$-space which are the convex hull of their profiles is described. Thus a generalization of Krein-Milman theorem\cite{Lay:1982} to a class of closed non-compact convex sets is obtained. Sufficient and necessary conditions for convexity, affinity and starsh…

The paper extends a classical result to positively curved homogeneous spaces.

Building upon ideas of Hironaka, Bierstone-Milman, Malgrange and others we generalize the inverse and implicit function theorem (in differential, analytic and algebraic setting) to sets of functions of larger multiplicities (or ideals). This allows one to describe singularities given by a finite set of generators or by…

New theorem shows nearly spherical manifolds can be mapped from spheres.

The Mahler volume of a centrally symmetric convex body K is defined as M(K)= (Vol K)(Vol K^dual). Mahler conjectured that this volume is minimized when K is a cube. We introduce the bottleneck conjecture, which stipulates that a certain convex body K^diamond subset K X K^dual has least volume when K is an ellipsoid. If…

The paper derives inequalities for eigenvalues and eigenfunction norms on manifolds.

Proves equivalence of local and global $L^p$-Brunn-Minkowski inequalities.

Non-negative curvature affects Markov chains' mixing and expansion properties.

Unified study of Brunn-Minkowski conjectures for log-concave measures.

Paper connects probability density cuts to graph theory eigenfunctions.

Klartag recently gave a beautiful alternative proof of the isoperimetric inequalities of Levy-Gromov, Bakry-Ledoux, Bayle and E. Milman on weighted Riemannian manifolds. Klartag's approach is based on a generalization of the localization method (so-called needle decompositions) in convex geometry, inspired also by opti…

Proves global $L_p$-Brunn-Minkowski inequality for specific $p$ range.

We investigate the distribution of eigenvalues of the weighted Laplacian on closed weighted Riemannian manifolds of nonnegative Bakry-Émery Ricci curvature. We derive some universal inequalities among eigenvalues of the weighted Laplacian on such manifolds. These inequalities are quantitative versions of the previous t…

The paper studies concentration of measure on manifolds with boundary, focusing on $1$-Lipschitz functions.

The study proves the existence and properties of isoperimetric clusters in Riemannian manifolds with bounded geometry.

Bounds on Steklov eigenvalues for manifolds with boundary.

We establish a version of the bottleneck conjecture, which in turn implies a partial solution to the Mahler conjecture on the product $v(K) = (\Vol K)(\Vol K^\circ)$ of the volume of a symmetric convex body $K \in \R^n$ and its polar body $K^\circ$. The Mahler conjecture asserts that the Mahler volume $v(K)$ is minimiz…

We prove that if $(X,\mathsf{d},\mathfrak{m})$ is a metric measure space with $\mathfrak{m}(X)=1$ having (in a synthetic sense) Ricci curvature bounded from below by $K>0$ and dimension bounded above by $N\in [1,\infty)$, then the classic Lévy-Gromov isoperimetric inequality (together with the recent sharpening counter…

Minimal Gaussian surface area is achieved by cones over a regular simplex for $m>3$ sets partitioning $R^n$.

The paper proves new inequalities for convex hypersurfaces using centro-affine geometry.

The paper proves uniqueness of solutions to curvature problems using various methods.