Metric Properties of Euclidean Buildings

Cops and Robber game in higher-dimensional manifolds with spherical and Euclidean metric

The recently introduced variation of the game of cops and robber is played on geodesic spaces. In this paper we establish some general strategies for the players, in particular the generalized radial strategy and the covering space strategy. Those strategies are then applied to the game on the $n$-dimensional ball, the sphere, and the torus.

Euclidean quotients of finite metric spaces

This paper is devoted to the study of quotients of finite metric spaces. The basic type of question we ask is: Given a finite metric space M, what is the largest quotient of (a subset of) M which well embeds into Hilbert space. We obtain asymptotically tight bounds for these questions, and prove that they exhibit phase transitions. We also study the analogous problem for embedings into l_p, and the particular case of the hypercube.

Extensions of isometric embeddings of Pseudo-Euclidean metric polyhedra

We extend the results of B. Minemyer by showing that any indefinite metric polyhedron (either compact or not) with the vertex degree bounded from above admits an isometric simplicial embedding into a Minkowski space of the lowest possible dimension. We provide a simple algorithm of constructing such embeddings. We also show that every partial simplicial isometric embedding of such space in general position extends to a simplicial isometric embedding of the whole space.

Isoperimetric inequalities of euclidean type in metric spaces

In this paper we prove an isoperimetric inequality of euclidean type for complete metric spaces admitting a cone-type inequality. These include all Banach spaces and all complete, simply-connected metric spaces of non-positive curvature in the sense of Alexandrov or, more generally, of Busemann. The main theorem generalizes results of Gromov and Ambrosio-Kirchheim.

A rigidity result for metric measure spaces with Euclidean heat kernel

We prove that a metric measure space equipped with a Dirichlet form admitting an Euclidean heat kernel is necessarily isometric to the Euclidean space. This helps us providing an alternative proof of Colding's celebrated almost rigidity volume theorem via a quantitative version of our main result. We also discuss the case of a metric measure space equipped with a Dirichlet form admitting a spherical heat kernel.

Metric characterizations of spherical, and Euclidean buildings

A building is a simplicial complex with a covering by Coxeter complexes (called apartments) satisfying certain combinatorial conditions. A building whose apartments are spherical (respectively Euclidean) Coxeter complexes has a natural piecewise spherical (respectively Euclidean) metric with nice geometric properties. We show that spherical and Euclidean buildings are completely characterized by some simple, geometric properties.

Bourgain-Brezis-Mironescu approach in metric spaces with Euclidean tangents

In the setting of metric measure spaces satisfying the doubling condition and the $(1,p)$-Poincaré inequality, we prove a metric analogue of the Bourgain-Brezis-Mironescu formula for functions in the Sobolev space $W^{1,p}(X,d,ν)$, under the assumption that for $ν$-a.e. point the tangent space in the Gromov-Hausdorff sense is Euclidean with fixed dimension $N$.

Spherical and hyperbolic conics

This is a survey of metric properties of non-Euclidean conics, mainly based on works of Chasles and Story. A spherical conic is the intersection of the sphere with a quadratic cone; similarly, a hyperbolic conic is the intersection of the Beltrami-Cayley-Klein disk with an affine conic. Non-Euclidean conics have metric properties similar to those of Euclidean conics, and even more due to the polarity that works here better than in the Euclidean plane.

Rigidity of Derivations in the Plane and in Metric Measure Spaces

Following Weaver we study generalized differential operators, called (metric) derivations, and their linear algebraic properties. In particular, for k = 1, 2 we show that measures on k-dimensional Euclidean space that induce rank-k modules of derivations must be absolutely continuous to Lebesgue measure. An analogous result holds true for measures concentrated on k-rectifiable sets with respect to k-dimensional Hausdorff measure. Though formulated for Euclidean spaces, these rigidity results also apply to the metric space setting and specifically, to spaces that support a doubling measure and …

An Upper Bound of the Total Q-Curvature and Its Isoperimetric Deficit for Higher-dimensional Conformal Euclidean Metrics

The aim of this paper is to give not only an explicit upper bound of the total Q-curvature but also an induced isoperimetric deficit formula for the complete conformal metrics on $\mathbb R^n$, $n\ge 3$ with scalar curvature being nonnegative near infinity and Q-curvature being absolutely convergent.

Does The Monge Theorem Apply To Some Non-Euclidean Geometries ?

In geometry, Monge's theorem states that for any three nonoverlapping circles of distinct radii in the two dimensional analytical plane equipped with the Euclidean metric, none of which is completely inside one of the others, the intersection points of each of the three pairs of external tangent lines are collinear. So, it is clearly observed that Monge's theorem is an application of Desargues' theorem. Our main motivation in this study is to show whether Monge theorem is still valid even if the plane is equipped with the metrics alpha and Lp.

Euclidean spaces as weak tangents of infinitesimally Hilbertian metric spaces with Ricci curvature bounded below

We show that in any infinitesimally Hilbertian $CD^*(K,N)$-space at almost every point there exists a Euclidean weak tangent, i.e. there exists a sequence of dilations of the space that converges to a Euclidean space in the pointed measured Gromov-Hausdorff topology. The proof follows by considering iterated tangents and the splitting theorem for infinitesimally Hilbertian $CD^*(0,N)$-spaces.

Proper quadrics in the Euclidean $n$-space

In this paper we investigate the metric properties of quadrics and cones of the $n$-dimensional Euclidean space. As applications of our formulas we give a more detailed description of the construction of Chasles and the wire model of Staude, respectively.

A simple proof of an isoperimetric inequality for euclidean and hyperbolic cone-surfaces

We prove that the isoperimetric inequalities in the euclidean and hyperbolic plane hold for all euclidean, respectively hyperbolic, cone-metrics on a disk with singularities of negative curvature. This is a discrete analog of the theorems of Weil and Bol that deal with Riemannian metrics of curvature bounded from above by 0, respectively by -1. A stronger discrete version was proved by A.D.Alexandrov, with a subsequent extension by approximation to metrics of bounded integral curvature. Our proof uses "discrete conformal deformations" of the metric that eliminate the singularities and increase…

Sublinear Higson corornae of Euclidean cones

The aim of this paper is to introduce the sublinear Higson corona and show that the sublinear Higson corona of Euclidean cone of P and X is decomposed into the product of P and that of X. Here P is a compact metric space and X is unbounded proper metric space. For example, the sublinear Higson corona of n-dimensional Euclidean space is homeomorphic to the product of (n-1)-dimensional sphere and that of natural numbers.

Undistorted fillings in subsets of metric spaces

We prove that if a quasiconvex subset $X$ of a metric space $Y$ has finite Nagata dimension and is Lipschitz $k$-connected or admits Euclidean isoperimetric inequalities up to dimension $k$ for some $k$ then $X$ is isoperimetrically undistorted in $Y$ up to dimension $k+1$. This generalizes and strengthens a recent result of the third named author and has several consequences and applications. It yields for example that in spaces of finite Nagata dimension, Lipschitz connectedness implies Euclidean isoperimetric inequalities, and Euclidean isoperimetric inequalities imply coning inequalities. …

The Funk and Hilbert geometries for spaces of constant curvature

The goal of this paper is to introduce and study analogues of the Euclidean Funk and Hilbert metrics on open convex subsets $Ω$ of hyperbolic or spherical spaces. At least at a formal level, there are striking similarities among the three cases: Euclidean, spherical and hyperbolic. We start by defining non-Euclidean analogues of the Euclidean Funk weak metric and we give three distinct representations of it in each of the non-Euclidean cases, which parallel the known situation for the Euclidean case. As a consequence, all of these metrics are shown to be Finslerian, and the associated norms of…

Bi-Lipschitz embedding of projective metrics

We give a sufficient condition for a projective metric on a subset of a Euclidean space to admit a bi-Lipschitz embedding into Euclidean space of the same dimension.

Metric thickenings of Euclidean submanifolds

Given a sample $Y$ from an unknown manifold $X$ embedded in Euclidean space, it is possible to recover the homology groups of $X$ by building a Vietoris--Rips or Čech simplicial complex on top of the vertex set $Y$. However, these simplicial complexes need not inherit the metric structure of the manifold, in particular when $Y$ is infinite. Indeed, a simplicial complex is not even metrizable if it is not locally finite. We instead consider metric thickenings, called the \emph{Vietoris--Rips} and \emph{Čech thickenings}, which are equipped with the 1-Wasserstein metric in place of the simplicia…

Uniformization of two-dimensional metric surfaces

We establish uniformization results for metric spaces that are homeomorphic to the euclidean plane or sphere and have locally finite Hausdorff 2-measure. Applying the geometric definition of quasiconformality, we give a necessary and sufficient condition for such spaces to be QC equivalent to the euclidean plane, disk, or sphere. Moreover, we show that if such a QC parametrization exists, then the dilatation can be bounded by 2. As an application, we show that the euclidean upper bound for measures of balls is a sufficient condition for the existence of a 2-QC parametrization. This result give…

Embedding fractals in Banach, Hilbert or Euclidean spaces

By a metric fractal we understand a compact metric space $K$ endowed with a finite family $\mathcal F$ of contracting self-maps of $K$ such that $K=\bigcup_{f\in\mathcal F}f(K)$. If $K$ is a subset of a metric space $X$ and each $f\in\mathcal F$ extends to a contracting self-map of $X$, then we say that $(K,\mathcal F)$ is a fractal in $X$. We prove that each metric fractal $(K,\mathcal F)$ is $\bullet$ isometrically equivalent to a fractal in the Banach spaces $C[0,1]$ and $\ell_\infty$; $\bullet$ bi-Lipschitz equivalent to a fractal in the Banach space $c_0$; $\bullet$ isometrically equivale…

Ulam的一个问题与de Rham定理

A question of Ulam and a theorem of de Rham

将de Rham分解定理推广至固有度量空间，给出唯一分解为欧几里得空间与不可约因子的ℓ²乘积，并部分回答Ulam的问题。

Two-dimensional metric spheres from gluing hemispheres

We study metric spheres Z obtained by gluing two hemispheres of the Euclidean sphere along an orientation-preserving homeomorphism mapping the equator onto itself, where the distance on Z is the canonical distance that is locally isometric to the spherical distance off the seam. We show that if Z is quasiconformally equivalent to the sphere, in the geometric sense, then g is a welding homeomorphism with conformally removable welding curves. We also show that g is bi-Lipschitz if and only if Z has a 1-quasiconformal parametrization whose Jacobian is comparable to the Jacobian of a quasiconforma…

Intrinsic isometries in Euclidean space

I consider compact metric spaces which admit intrinsic isometries to Euclidean d-space. The main result roughly states that the class of these spaces coincides with class of inverse limits of Euclidean d-polyhedra.

Lipschitz and path isometric embeddings of metric spaces

We prove that each sub-Riemannian manifold can be embedded in some Euclidean space preserving the length of all the curves in the manifold. The result is an extension of Nash $C^1$ Embedding Theorem. For more general metric spaces the same result is false, e.g., for Finsler non-Riemannian manifolds. However, we also show that any metric space of finite packing dimension can be embedded in some Euclidean space via a Lipschitz map.

欧几里得集的幅度同调等价

Magnitude homology equivalence of Euclidean sets

本文研究了欧几里得集在幅度同调意义上的等价条件，引入了内边界和核心的概念，并加强了Carathéodory定理。