Related papers: On the Gomori-Hu inequality

(Write-up of a talk at the Bialowieza meeting, July 2007.) Gelfand and Kolmogorov in 1939 proved that a compact Hausdorff topological space $X$ can be canonically embedded into the infinite-dimensional vector space $C(X)^* $, the dual space…

Let (X,dX) and (Y,dY) be semimetric spaces with distance sets D(X) and, respectively, D(Y). A mapping F : X \to Y is a weak similarity if it is surjective and there exists a strictly increasing f : D(Y) \to D(X) such that dX = f \circ dY…

Recently, much interest was devoted to the Urysohn universal metric space U and its isometries; this paper is a contribution to this field of research. In particular, we study some properties of isometries of U, and prove the following…

We prove an inequality for submanifolds of Cartan-Hadamard manifolds, which relates the geometry of a submanifold to the measure of the geodesics in the ambient space which it intersects. For hypersurfaces, this gives an extension of…

For each given $p\in[1,\infty]$ we investigate certain sub-family $\mathcal{M}_p$ of the collection of all compact metric spaces $\mathcal{M}$ which are characterized by the satisfaction of a strengthened form of the triangle inequality…

The inequality $|X| \leq 2^{\chi(X)}$ has been proved to be true for Lindel\"of spaces (Arhangel'ski\u\i, 1969), $H$-closed spaces (Dow-Porter, 1982) and ccc spaces (Hajnal-Ju\'asz 1967), by quite different arguments. We present a common…

We show that there are homotopy equivalences $h:N\to M$ between closed manifolds which are induced by cell-like maps $p:N\to X$ and $q:M\to X$ but which are not homotopic to homeomorphisms. The phenomenon is based on construction of…

In a recent paper A. Cianchi, N. Fusco, F. Maggi, and A. Pratelli have shown that, in the Gauss space, a set of given measure and almost minimal Gauss boundary measure is necessarily close to be a half-space. Using only geometric tools, we…

The modulus metric (also called the capacity metric) on a domain $D\subset \mathbb{R}^n$ can be defined as $\mu_D(x,y)=\inf\{{\mbox{cap}}\,(D,\gamma)\}$, where ${\mbox{cap}}\,(D,\gamma)$ stands for the capacity of the condenser $(D,\gamma)$…

Gomory and Hu proved that if $ G $ is a finite graph with non-negative weights on its edges, then there exists a tree $ T $ (called now Gomory-Hu tree) on $ V(G) $ such that for all $ u\neq v\in V(G) $ there is an $ e\in E(T) $ such that…

The set $dd(X)$ of densities of all dense subspaces of a topological space $X$ is called the double density spectrum of $X$. In this note we present a couple of results that imply $\lambda \in dd(X)$, provided that $X$ is a compact space…

Let $V$ be a Banach space where for fixed $n$, $1<n<\dim(V)$, all of its $n$-dimensional subspaces are isometric. In 1932, Banach asked if under this hypothesis $V$ is necessarily a Hilbert space. Gromov, in 1967, answered it positively for…

It is proved that the Gromov-Hausdorff metric on the space of compact metric spaces considered up to an isometry is strictly intrinsic, i.e., the corresponding metric space is geodesic. In other words, each two points of this space (each…

Let $\Pi$ be a polar space of type $\textsf{D}_{n}$. Denote by ${\mathcal G}_{\delta}(\Pi)$, $\delta\in \{+,-\}$ the associated half-spin Grassmannians and write $\Gamma_{\delta}(\Pi)$ for the corresponding half-spin Grassmann graphs. In…

For any intrinsic Gromov hyperbolic space we establish a Gehring-Hayman type theorem for conformally deformed spaces. As an application, we prove that any complete intrinsic hyperbolic space with atleast two points in the Gromov boundary…

The relation between negatively curved spaces and their boundaries is important for various rigidity problems. In \cite{biswas2024quasi}, the class of Gromov hyperbolic spaces called maximal Gromov hyperbolic spaces was introduced, and the…

We show that the set of k-dimensional isoperimetric exponents of finitely presented groups is dense in the interval [1, \infty) for k > 1. Hence there is no higher-dimensional analogue of Gromov's gap (1,2) in the isoperimetric spectrum.

We prove, using elementary methods of complex analysis, the following generalization of the isoperimetric inequality: if $p\in\re$, $\Omega\subset\re^2$ then the inequality $$…

Let $X$ be a compact connected Riemann surface of genus $g$, with $g\geq 2$, and ${\cal M}_{\xi}$ a smooth moduli space of fixed determinant semistable vector bundles of rank $n$, with $n\geq 2$, over $X$. Take a smooth anticanonical…

We find many examples of compact Riemannian manifolds $(M,g)$ whose closed minimal hypersurfaces satisfy a lower bound on their index that is linear in their first Betti number. Moreover, we show that these bounds remain valid when the…