度量几何

Let $D\subsetneq\mathbb{R}^n,~n\ge 2$, be a domain. In this manuscript, a new version of the Vuorinen's distance ratio metric $j_D$ [{\tt J. Analyse Math.} {\bf 45} (1985), 69--115], denoted by $\zeta_D$, and a version of Gehring-Osgood's…

Finding a largest Euclidean ball in a given convex body $K \subset \mathbb{R}^d$ and finding a largest volume ellipsoid in $K$ are two problems of fundamentally different nature. The first is a purely Euclidean problem, where we consider…

We prove generalized versions of the Variance Inequality known for barycenters in CAT(0) spaces, inspired by an analogous result for $p$-uniformly convex Banach spaces. Our generalizations apply to balls of sufficiently small radius in…

Recently, Yang, Yuan and Zhang [Doubling properties of self-similar measures and Bernoulli measures on self-affine Sierpinski sponges, Indiana Univ. Math. J., 73 (2024), 475-492] characterized when a self-similar measure satisfying the open…

We introduce a type of surgery on metric spaces. This surgery, in some sense, seeks to replace a subspace $S$ of a metric space $X$ with another metric space $T$ via a function $f : S \to T$. When $T$ is a discrete space, this amounts to…

Although the hyperbolic metric possesses many remarkable properties, it is not defined on arbitrary subdomains of $\mathbb{R}^n$ with $n \geq 2$. This article introduces a new hyperbolic-type metric that provides an alternative approach to…

Nonlinear Poincar\'e inequalities are indispensable tools in the study of dimension reduction and low-distortion embeddings of graphs into metric spaces, and have found remarkable algorithmic applications. A basic open problem, posed by Jon…

We show that every real-valued Lipschitz function on a subset of a metric space can be extended to the whole space while preserving the slope and, up to a small error, the global Lipschitz constant. This answers a question posed by Di…

We address an old open question in convex geometry that dates back to the work of Minkowski: what are the equality cases of the monotonicity of mixed volumes? The problem is equivalent to that of providing a geometric characterization of…

We establish geometric relationships between the average scale-invariant Cassinian metric and other hyperbolic type metrics. In addition, we study the local convexity properties of the scale-invariant metric balls in Euclidean once…

We prove that for each positive integer $n$, the Rips complexes of the $n$-dimensional integer lattice in the $d_1$ metric (i.e., the Manhattan metric, also called the natural word metric in the Cayley graph) are contractible at scales…

Nonabelian simply connected nilpotent Lie groups and not virtually abelian finitely generated groups of polynomial growth do not quasi-isometrically embed into uniformly convex Banach spaces. We quantify this fact by showing that a ball of…

The Cohn-Elkies linear program for sphere packing, which was used to solve the 8 and 24 dimensional cases, is conjectured to not be sharp in any other dimension $d>2$. By mapping feasible points of this infinite-dimensional linear program…

We establish an equivalence between the rigidity of Wasserstein contraction along heat flows and the rigidity of Bakry--\'Emery gradient estimates for Lipschitz functions. Applying results of Ambrosio--Bru\'e--Semola and Han, we show that…

In this note, we present examples showing that several natural ways of constructing lattices from error-correcting codes do not in general yield a correspondence between minimum-weight non-zero codewords and shortest non-zero lattice…

In this paper we prove that the $\ell_0$ isoperimetric coefficient for any axis-aligned cubes, $\psi_{\mathcal{C}}$, is $\Theta(n^{-1/2})$ and that the isoperimetric coefficient for any measurable body $K$, $\psi_K$, is of order…

Chromatic metric pairs consist of a metric space and a coloring function partitioning a subset thereof into various colors. It is a natural extension of the notion of chromatic point sets studied in chromatic topological data analysis. A…

The discrete Laplacian on Euclidean triangulated surfaces is a well-established notion. We introduce discrete Laplacians on spherical and hyperbolic triangulated surfaces. On the one hand, our definitions are close to the Euclidean one in…

In this paper, we give a short and self-contained proof to a 1991 conjecture by Moore concerning the structure of certain finite-dimensional Gromov--Hausdorff limits, in the ANR setting. As a consequence, one easily characterizes finite…

A $\textit{spherical $t$-design curve}$ was defined by Ehler and Gr\"{o}chenig to be a continuous, piecewise smooth, closed curve on the sphere with finitely many self-intersections whose associated line integral applied to any polynomial…