Mathematics

Recently Kloeckner described the structure of the isometry group of the quadratic Wasserstein space W 2 ( R n ) . It turned out that the case of the real line is exceptional in the sense that there exists an exotic isometry flow. Following this line of investigation, we compute Isom( W p (R)) , the isometry group of the Wasserstein space W p (R) for all p∈[1,∞)∖{2} . We show that W 2 (R) is also exceptional regarding the parameter p : W p (R) is isometrically rigid if and only if p≠2 . Regarding the underlying space, we prove that the exceptionality of p=2 disappears if we replace R by the compact interval [0,1] . Surprisingly, in that case, W p ([0,1]) is isometrically rigid if and only if p≠1 . Moreover, W 1 ([0,1]) admits isometries that split mass, and Isom( W 1 ([0,1])) cannot be embedded into Isom( W 1 (R)).

We study the conditions under which the isometry of spaces with metrics generated by weights given on the edges of finite trees is equivalent to the isomorphism of these trees. Similar questions are studied for ultrametric spaces generated by labelings given on the vertices of trees. The obtained results generalized some facts previously known for phylogenetic trees and for Gurvich-Vyalyi monotone trees.

The classical isoperimetric inequality can be extended to a general normed plane. In the Euclidean plane, the defect in the isoperimetric inequality can be calculated in terms of the signed areas of some singular sets. In this paper we consider normed planes with smooth by parts unit balls and the corresponding class of admissible curves. For such an admissible curve, the singular sets are defined as projections in the subspaces of symmetric and constant width admissible curves. In this context, we obtain some improved isoperimetric inequalities whose equality hold for symmetric or constant width curves.

In this paper, we investigate eigenvalues of the Wentzel-Laplace operator on a bounded domain in some Riemannian manifold. We prove asymptotically optimal estimates, according to the Weyl's law through bounds that are given in terms of the isoperimetric ratio of the domain. Our results show that the isoperimetric ratio allows to control the entire spectrum of the Wentzel-Laplace operator in various ambient spaces.

The aim of this note is to investigate isoperimetric-type problems for 3-dimensional parallelohedra; that is, for convex polyhedra whose translates tile the 3-dimensional Euclidean space. Our main result states that among 3-dimensional parallelohedra with unit volume the one with minimal mean width is the regular truncated octahedron. In addition, we establish a connection between the edge lengths of 3-dimensional parallelohedra and the edge densities of the translative mosaics generated by them, and use our method to prove that among translative, convex mosaics generated by a parallelohedron with a given volume, the one with minimal edge density is the face-to-face mosaic generated by cubes.

Let (M,d) be a separable and complete geodesic space with curvature lower bounded, by κ∈R , in the sense of Alexandrov. Let μ be a Borel probability measure on M , such that μ∈ P 2 (M) , and that has at least one barycenter x ∗ ∈M . We show that for any geodesically α -convex function f:M→R , for α∈R , the inequality f( x ∗ )≤ ∫ M (f− α 2 d 2 ( x ∗ ,.))dμ, holds provided f is locally Lipschitz at x ∗ and either positive or in L 1 (μ) . Our proof relies on the properties of tangent cones at barycenters and on the existence of gradients for semi-concave functions in spaces with lower bounded curvature.

Very recently J. Kotrbaty has proven general inequalities for translation invariant smooth valuations formally analogous to the Hodge- Riemann bilinear relations in the Kahler geometry. The goal of this note is to apply Kotrbaty's theorem to obtain a few apparently new inequalities for mixed volumes of convex bodies.

In this paper, we consider the Monge optimal transport problem with distance cost. We prove that in some metric spaces, possibly with many branching geodesics, an optimal transport map exists if the first marginal is absolutely continuous. The result is applicable to normed spaces and Hilbert geometries.

We study the following question: for given d?? , n?�d and k?�n , what is the largest value c(d,n,k) such that from any set of n unit vectors in R d , we may select k vectors with corresponding signs ±1 so that their signed sum has norm at least c(d,n,k) ? The problem is dual to classical vector sum minimization and balancing questions, which have been studied for over a century. We give asymptotically sharp estimates for c(d,n,k) in the general case. In several special cases, we provide stronger estimates: the quantity c(d,n,n) corresponds to the ??p -polarization problem, while determining c(d,n,2) is equivalent to estimating the coherence of a vector system, which is a special case of p -frame energies. Two new proofs are presented for the classical Welch bound when n=d+1 . For large values of n , volumetric estimates are applied for obtaining fine estimates on c(d,n,2) . Studying the planar case, sharp bounds on c(2,n,k) are given. Finally, we determine the exact value of c(d,d+1,d+1) under some extra assumptions.

The course was given at Peking University, Fall 2019. We discuss the following subjects: (1) Introduction to general topology, hyperspaces, metric and pseudometric spaces, graph theory. (2) Graphs in metric spaces, minimum spanning tree, Steiner minimal tree, Gromov minimal filling. (3) Hausdorff distance, Vietoris topology, Limits theory, inheritance of completeness, total boundedness, compactness by hyperspaces. (4) Gromov-Hausdorff distance, triangle inequality, positive definiteness for isometry classes of compact spaces, counterexample for boundedly compact spaces. (5) Gromov-Hausdorff distance for separable spaces in terms of their isometric images in \ell_\infty, correspondences, Gromov-Hausdorff distance in terms of correspondences. (6) Epsilon-isometries and Gromov-Hausdorff distance. (7) Irreducible correspondences and Gromov-Hausdorff distance. (8) Gromov-Hausdorff convergence, inheritance of metric and topological properties while Gromov-Hausdorff convergence. (9) Gromov-Hausdorff space (GH-space), optimal correspondences, existence of closed optimal correspondences for compact metric spaces, GH-space is geodesic. (10) Cover number, packing number, total boundedness, completeness, and separability of GH-space. (11) mst-spectrum in terms of GH-distances to simplexes, Steiner problem in GH-space. (12) GH-distance to simplexes with more points, GH-distance to simplexes with at most the same number of points. (13) Generalized Borsuk problem, solution of Generalized Borsuk problem in terms of GH-distances, clique covering number and chromatic number of simple graphs, their dualities, calculating these numbers in terms of GH-distances.