arXiv research

Study shows intrinsic timed Hausdorff convergence leads to Gromov-Hausdorff and big bang convergence.

Upper bound for Hausdorff distance between hyperbolic space and its medianization.

Study uses equivariant topology to measure distances between G metric spaces.

Proves compactness for timed-metric spaces using new distance and maps.

Compactness theorem for timed-metric spaces established.

Research shows surfaces close to planes in Hausdorff distance.

Gromov-Hausdorff distances measure shape difference between the objects representable as compact metric spaces, e.g. point clouds, manifolds, or graphs. Computing any Gromov-Hausdorff distance is equivalent to solving an NP-Hard optimization problem, deeming the notion impractical for applications. In this paper we pro…

This paper introduces a new distance metric for filtered A-infinity categories, focusing on Lagrangian submanifolds.

Proves Hölder-type inequality for Lagrangians' distance.

The paper connects geometric and topological concepts to bound distances between metric spaces.

In the present paper we calculate the Gromov-Hausdorff distance between an arbitrary simplex (a metric space all whose non-zero distances are the same) and a finite metric space whose non-zero distances take two distinct values (so-called $2$-distance spaces). As a corollary, a complete solution to generalized Borsuk p…

We introduce an asymmetric distance function, which we call the `left Hausdorff distance function', on the space of geodesic laminations on a closed hyperbolic surface of genus at least 2. This distance is an asymmetric version of the Hausdorff distance between compact subsets of a metric space. We prove a rigidity res…

In this paper, we discuss how a Gromov-Hausdorff-like distance function over the space of all isometric classes of compact $C^k$-Riemannian manifolds should be defined in the aspect of the Riemannan submanifold theory, where $k\geq 1$. The most important fact in this discussion is as follows. The Hausdorff distance fun…

One of the most beautiful notions of metric geometry is the Gromov-Hausdorff distance which measures the difference between two metric spaces. To define the distance, let us isometrically embed these spaces into various metric spaces and measure the Hausdorff distance between their images. The best matching corresponds…

The paper explores null distance convergence for warped product spacetimes.

We investigate compact Hausdorff foliations on compact Riemannian manifolds in the context of the Gromov-Hausdorff distance theory. We give some sufficient conditions for such foliations to be separated in the Gromov-Hausdorff topology.

Circle's metric is at least π/4 away from any simply connected geodesic space.

Bounds and constructions for Gromov-Hausdorff distance between spheres.

Total curvatures of certain hypersurfaces are continuous.

A clustering procedure, based on the Hausdorff distance, is introduced and tested on the financial time series of the Dow Jones Industrial Average (DJIA) index.

Study extends null distance concept to Lorentzian length spaces for spacetime analysis.

The Gromov-Hausdorff distance provides a metric on the set of isometry classes of compact metric spaces. Unfortunately, computing this metric directly is believed to be computationally intractable. Motivated by applications in shape matching and point-cloud comparison, we study a semidefinite programming relaxation of …

Two singular mean curvature flows converge if Hausdorff distance decreases faster than inverse time.

Defines new metrics for Lorentzian spaces and their convergence.

The paper converts metric bounds to distance function Hölder bounds and proves compactness theorems.

The paper constructs Gromov-Hausdorff metrics for Lorentzian spaces and calculates dimensions.

Quantifies closeness of special Lagrangians under Floer conditions.

We find lower and upper bounds for the risk of estimating a manifold in Hausdorff distance under several models. We also show that there are close connections between manifold estimation and the problem of deconvolving a singular measure.

A clustering algorithm based on the Hausdorff distance is introduced and compared to the single and complete linkage. The three clustering procedures are applied to a toy example and to the time series of financial data. The dendrograms are scrutinized and their features confronted. The Hausdorff linkage relies of firm…

New method uses cohomology to quantify molecular similarity.

The paper studies convergence of cosmological spacetimes using null distance.

In this paper we consider the problem of minimizing the relative perimeter under a volume constraint in the interior of a convex body, i.e., a compact convex set in Euclidean space with interior points. We shall not impose any regularity assumption on the boundary of the convex set. Amongst other results, we shall prov…

On a complete, connected, locally compact, non-compact geodesic space $(X,d)$, we assign each compact set a distance-like function. With the help of these functions, we obtain a pseudo-metric on the space of (non-empty) compact subsets of $X$ which is less than the Hausdorff distance. The quotient metric space is close…

We show that if a closed surface in $\mathbb{R}^3$ has entropy near to that of the unit two-sphere, then the surface is close to a round two-sphere in the Hausdorff distance.

We show that if the entropy of any closed hypersurface is close to that of a round hyper-sphere, then it is close to a round sphere in Hausdorff distance. Generalizing the result of \cite{BW1} to higher dimensions.

By Gromov's compactness theorem for metric spaces, every uniformly compact sequence of metric spaces admits an isometric embedding into a common compact metric space in which a subsequence converges with respect to the Hausdorff distance. Working in the class or oriented $k$-dimensional Riemannian manifolds (with bound…

Study horocycle orbits in $\mathbb{Z}$-covers of hyperbolic surfaces.

Study proves inequality for hypersurfaces and shows almost extremals are close to Wulff shape.

We show that the Hausdorff distance between any forward and any backward surgery paths in the sphere graph is at most 2. From this it follows that the Hausdorff distance between any two surgery paths with the same initial sphere system and same target sphere system is at most 4. Our proof relies on understanding how su…

Uniform foliations with Reeb components on 3-manifolds.

Proves a theorem for Assouad dimension with applications to distance sets and radial projections.

We continue our investigation of the space of geodesic laminations on a surface, endowed with the Hausdorff topology. We determine the topology of this space for the once-punctured torus and the 4-times-punctured sphere. For these two surfaces, we also compute the Hausdorff dimension of the space of geodesic lamination…

We construct a compact metric space that has any other compact metric space as a tangent, with respect to the Gromov-Hausdorff distance, at all points. Furthermore, we give examples of compact sets in the Euclidean unit cube, that have almost any other compact set of the cube as a tangent at all points or just in a den…

Let $M$ be a compact Riemannian manifold with boundary. We show that $M$ is Gromov-Hausdorff close to a convex Euclidean region $D$ of the same dimension if the boundary distance function of $M$ is $C^1$-close to that of $D$. More generally, we prove the same result under the assumptions that the boundary distance func…

Close to complex projective spaces, Ricci shrinkers are rigid.

Given a geodesic space (E, d), we show that full ordinal knowledge on the metric d-i.e. knowledge of the function D d : (w, x, y, z) $\rightarrow$ 1 d(w,x)$\le$d(y,z) , determines uniquely-up to a constant factor-the metric d. For a subspace En of n points of E, converging in Hausdorff distance to E, we construct a met…

Generalizes Connes's tangent groupoid for sub-Riemannian geometry.

The paper examines how nonlinear transformations affect ridge sets in manifold learning.