Mathematics - Metric Geometry Publications (50)


Mathematics - Metric Geometry Publications

Let $(\mathbb{X} , d, \mu )$ be a proper metric measure space and let $\Omega \subset \mathbb{X}$ be a bounded domain. For each $x\in \Omega$, we choose a radius $0< \varrho (x) \leq \mathrm{dist}(x, \partial \Omega ) $ and let $B_x$ be the closed ball centered at $x$ with radius $\varrho (x)$. If $\alpha \in \mathbb{R}$, consider the following operator in $C( \overline{\Omega} )$, $$ \mathcal{T}_{\alpha}u(x)=\frac{\alpha}{2}\left(\sup_{B_x } u+\inf_{B_x } u\right)+(1-\alpha)\,\frac{1}{\mu(B_x)}\int_{B_x}\hspace{-0. Read More

The aim of this note is to announce some results about the probabilistic and deterministic asymptotic properties of linear groups. The first one is the analogue, for norms of random matrix products, of the classical theorem of Cramer on large deviation principles (LDP) for sums of iid real random variables. In the second result, we introduce a limit set describing the asymptotic shape of the powers of a subset S of a semisimple linear Lie group G (e. Read More

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. Read More

In this paper, we are going to discuss the following problem: Let $T$ be a fixed set in $\mathbb{R}^n$. And let $S$ and $B$ he two subsets in $\mathbb{R}^n$ such that for any $x$ in $S$, there exists an $r$ such that $x+ r T$ is a subset of $B$. How small can be $B$ be if we know the size of $S$? Stein proved that for $n$ is greater than or equal to 3 and $T$ is a sphere centered at origin, then $S$ has positive measure implies $B$ has positive measure using spherical maximal operator. Read More

We discuss the notion of sublinearly bilipschitz maps, which generalize quasi-isometries, allowing some additional terms that behave sublinearly with respect to the distance to the origin. Such maps were originally motivated by the fact they induce bilipschitz homeomorphisms between asymptotic cones. We prove here that for hyperbolic groups, they also induce H\"older homeomorphisms between the boundaries. Read More

We use the language of precategories to formulate a general mathematical framework for phylogenetics. Read More

We obtain a topological and equivariant classification of closed, connected three-dimensional Alexandrov spaces admitting a local isometric circle action. We show, in particular, that such spaces are homeomorphic to connected sums of some closed 3-manifold with a local circle action and finitely many copies of the suspension of the real projective plane. Read More

In a $d$-dimensional convex body $K$, for $n \leq d+1$, random points $X_0, \dots, X_{n-1}$ are chosen according to the uniform distribution in $K$. Their convex hull is a random $(n-1)$-simplex with probability $1$. We denote its $(n-1)$-dimensional volume by $V_{K[n]}$. Read More

It is unknown if there exists a locally $\alpha$-H\"older homeomorphism $f:\mathbb{R}^3\to \mathbb{H}^1$ for any $\frac{1}{2}< \alpha\le \frac{2}{3}$, although the identity map $\mathbb{R}^3\to \mathbb{H}^1$ is locally $\frac{1}{2}$-H\"older. More generally, Gromov asked: Given $k$ and a Carnot group $G$, for which $\alpha$ does there exist a locally $\alpha$-H\"older homeomorphism $f:\mathbb{R}^k\to G$? Here, we equip a Carnot group $G$ with the Carnot-Carath\'eodory metric. In 2014, Balogh, Hajlasz, and Wildrick considered a variant of this problem. Read More

Alexandrov's inequalities imply that for any convex body $A$, the sequence of intrinsic volumes $V_1(A),\ldots,V_n(A)$ is non-increasing (when suitably normalized). Milman's random version of Dvoretzky's theorem shows that a large initial segment of this sequence is essentially constant, up to a critical parameter called the Dvoretzky number. We show that this near-constant behavior actually extends further, up to a different parameter associated with $A$. Read More

We provide a unified approach to the existence, uniqueness and interior regularity of solutions to the Dirichlet problem of Korevaar and Schoen in the setting of mappings between singular metric spaces. More precisely, under mild conditions on the metric spaces $X$ and $Y$, we obtain the existence of solutions for the Dirichlet problem of Korevaar and Schoen. When $Y$ has non-positive curvature in the sense of Alexandrov (NPC), solutions are shown to be unique and local H\"older continuous. Read More

We relate $L^{q,p}$-cohomology of bounded geometry Riemannian manifolds to a purely metric space notion of $\ell^{q,p}$-cohomology, packing cohomology. This implies quasi-isometry invariance of $L^{q,p}$-cohomology together with its multiplicative structure. The result partially extends to the Rumin $L^{q,p}$-cohomology of bounded geometry contact manifolds. Read More

We describe our initial explorations in simulating non-euclidean geometries in virtual reality. Our simulation of the product of two-dimensional hyperbolic space with one-dimensional euclidean space is available at http://h2xe.hypernom. Read More

With a view to establishing measure theoretic approximation properties of Delone sets, we study a setup which arises naturally in the problem of averaging almost periodic functions along exponential sequences. In this setting, we establish a full converse of the Borel-Cantelli lemma. This provides an analogue of more classical problems in the metric theory of Diophantine approximation, but with the distance to the nearest integer function replaced by distance to an arbitrary Delone set. Read More

In \cite{LiWang2001complete1,LiWang2001complete2}, Li-Wang proved a splitting theorem for an n-dimensional Riemannian manifold with $Ric\geqslant -(n-1)$ and the bottom of spectrum $\lambda_0(M)=\frac{(n-1)^2}{4}$. For an n-dimensional compact manifold $M$ with $Ric\geqslant-(n-1)$ with the volume entropy $h(M)=n-1$, Ledrappier-Wang \cite{LeW2010volent} proved that the universal cover $\tilde{M}$ is isometric to the hyperbolic space $\mathbb{H}^n$. We will prove analogue theorems for Alexandrov spaces. Read More

We prove that H-type Carnot groups of rank $k$ and dimension $n$ satisfy the $\mathrm{MCP}(K,N)$ if and only if $K\leq 0$ and $N \geq k+3(n-k)$. The latter integer coincides with the geodesic dimension of the Carnot group. The same result holds true for the larger class of generalized H-type Carnot groups introduced in this paper, and for which we compute explicitly the optimal synthesis. Read More

We investigate Minkowski additive, continuous, and translation invariant operators $\Phi:\mathcal{K}^n\to\mathcal{K}^n$ defined on the family of convex bodies such that the volume of the image $\Phi(K)$ is bounded from above and below by multiples of the volume of the convex body $K$, uniformly in $K$. We obtain a representation result for an infinite subcone contained in the cone formed by this type of operators. Under the additional assumption of monotonicity or $SO(n)$-equivariance, we obtain new characterization results for the difference body operator. Read More

We show that there is a sequence of subsets of each discrete Heisenberg group for which the non-singular ergodic theorem holds. The sequence depends only on the group; it works for any of its non-singular actions. To do this we use a metric which was recently shown by Le Donne and Rigot to have the Besicovitch covering property and then apply an adaptation of Hochman's proof of the multiparameter non-singular ergodic theorem. Read More

We study the change in the extrinsic geometry of a triangulated surface under infinitesimal conformal deformations in Euclidean space. A deformation of vertices is conformal if it preserves length cross-ratios. On the one hand, conformal deformations generalize deformations preserving edge lengths. Read More

Internal diffusion limited aggregation (IDLA) is a stochastic growth model on a graph $G$ which describes the formation of a random set of vertices growing from the origin (some fixed vertex) of $G$. Particles start at the origin of $G$ and perform simple random walks; each particle moves until it lands on a site which was not previously visited by any other particle. This random set of occupied sites in $G$ is called the IDLA cluster. Read More

We prove that for every planar centrally symmetric convex body $K$, among all ellipses $E$ with the same center and the same area as $K$, there exists a unique one for which the area of the symmetric difference $K \vartriangle E$ is minimal. We obtain this result by showing that this area is a strictly quasiconvex function of the ellipse $E$, with respect to a natural affine structure on the set of ellipses. Read More

We introduce an obstruction for the existence of a coarse embedding of a given group or space into a hyperbolic group, or more generally into a hyperbolic graph of bounded degree. The condition we consider is "admitting exponentially many fat bigons", and it is preserved by a coarse embedding between graphs with bounded degree. Groups with exponential growth and linear divergence (such as direct products of two groups one of which has exponential growth, solvable groups that are not virtually nilpotent, and uniform higher-rank lattices) have this property and hyperbolic graphs do not, so the former cannot be coarsely embedded into the latter. Read More

This work is related to billiards and their applications in geometric optics. It is known that perfectly invisible bodies with mirror surface do not exist. It is natural to search for bodies that are, in a sense, close to invisible. Read More

We consider a point cloud $X_n := \{ x_1, \dots, x_n \}$ uniformly distributed on the flat torus $\mathbb{T}^d : = \mathbb{R}^d / \mathbb{Z}^d $, and construct a geometric graph on the cloud by connecting points that are within distance $\epsilon$ of each other. We let $\mathcal{P}(X_n)$ be the space of probability measures on $X_n$ and endow it with a discrete Wasserstein distance $W_n$ as defined by Maas. We show that as long as $\epsilon= \epsilon_n$ decays towards zero slower than an explicit rate depending on the level of uniformity of $X_n$, then the space $(\mathcal{P}(X_n), W_n)$ converges in the Gromov-Hausdorff sense towards the space of probability measures on $\mathbb{T}^d$ endowed with the Wasserstein distance. Read More

We investigate a geometric structure on ${\rm Sym}^+(p)$, the set of $p \times p$ symmetric positive-definite matrices, based on eigen-decomposition. Eigenstructure determines both a stratification of ${\rm Sym}^+(p)$, defined by eigenvalue multiplicities, and fibers of the "eigen-composition" map $F:M(p):=SO(p)\times{\rm Diag}^+(p)\to{\rm Sym}^+(p)$. The fiber structure leads to the notions of {\em scaling-rotation distance} between $X,Y\in {\rm Sym}^+(p)$, the distance in $M(p)$ between fibers $F^{-1}(X)$ and $F^{-1}(Y)$, and {\em minimal smooth scaling-rotation (MSSR) curves} [Jung et al. Read More

We define a class of metric spaces we call manifold-like. This class includes Riemannian manifolds, metric graphs, products and some quotients of these as well as a number of more singular spaces. There exists a natural Dirichlet form based on the Laplacian on these spaces. Read More

We seek conditions under which colorings of various vector spaces are guaranteed to have a copy of a unit equilateral triangle, having each vertex in a different color class. In particular, we explore the analogous question in the setting of vector spaces finite fields, with an appropriate notion of distance. Read More

Let Z be the zero set of a holomorphic map f from C^n to C^k with f(0) = 0. We prove that for any r > 0, the Gaussian measure of the Euclidean r-neighborhood of Z is at least as large as the Gaussian measure of the Euclidean r-neighborhood of an (n-k)-dimensional complex subspace of C^n. Read More

We prove the strong Novikov conjecture for groups having polynomially bounded higher-order combinatorial functions. This includes all automatic groups. Read More

Until now, little was known about properties of small cells in a Poisson hyperplane tessellation. The few existing results were either heuristic or applying only to the two dimensional case and for really specific size measurements and directional distributions. This paper fills this gap by providing a systematic study of small cells in a Poisson hyperplane tessellation of arbitrary dimension, arbitrary directional distribution $\varphi$ and with respect to an arbitrary size measurement $\Sigma$. Read More

We construct classifying spaces for discrete and compact Lie groups, with the property that they are topological groups and complete metric spaces in a natural way. We sketch a program in view of extending these constructions. Read More

Short and transparent proofs of upper variance bounds and central limit theorems for intrinsic volumes of random polytopes in smooth convex bodies are presented. They combine different tools such as estimates for floating bodies with Stein's method from probability theory. Read More

In this article we study the role of the Green function for the Laplacian in a compact Riemannian manifold as a tool for obtaining well-distributed points. In particular, we prove that a sequence of minimizers for the Green energy is asymptotically uniformly distributed. We pay special attention to the case of locally harmonic manifolds. Read More

Let $n$ be a sufficiently large natural number and let $B$ be an origin-symmetric convex body in $R^n$ in the $\ell$-position, and such that the normed space $(R^n,\|\cdot\|_B)$ admits a $1$-unconditional basis. Then for any $\varepsilon\in(0,1/2]$, and for random $c\varepsilon\log n/\log\frac{1}{\varepsilon}$-dimensional subspace $E$ distributed according to the rotation-invariant (Haar) measure, the section $B\cap E$ is $(1+\varepsilon)$-Euclidean with probability close to one. This shows that the "worst-case" dependence on $\varepsilon$ in the randomized Dvoretzky theorem in the $\ell$-position is significantly better than in John's position. Read More

We prove that on a large family of metric measure spaces, if the `carr\'e du champ' $\Gamma$ satisfies a $p$-gradient estimate of heat flows for some $p>2$, then the metric measure space is ${\rm RCD}(K,\infty)$. The argument relies on the non-smooth Bakry-\'Emery's theory. As an application, we provide another proof of the von Renesse-Sturm's theorem on smooth metric measure space. Read More

We propose a general framework to study constructions of Euclidean lattices from linear codes over finite fields. In particular, we prove general conditions for an ensemble constructed using linear codes to contain dense lattices (i.e. Read More

Suppose that the $d$-dimensional unit cube $Q$ is the union of three disjoint "simple" sets $E$, $F$ and $G$ and that the volumes of $E$ and $F$ are both greater than half the volume of $G$. Does this imply that, for some cube $W$ contained in $Q$. the volumes of $E\cap W$ and $F\cap W$ both exceed $s$ times the volume of $W$ for some absolute positive constant $s$? Here, by "simple" we mean a set which is a union of finitely many dyadic cubes. Read More

This article proves the Voronoi conjecture on parallelotopes in the special case of 3-irreducible tilings. Parallelotopes are convex polytopes which tile the Euclidean space by their translated copies, like in the honeycomb arrangement of hexagons in the plane. An important example of parallelotope is the Dirichlet-Voronoi domain for a translation lattice. Read More

An infinitely smooth convex body in $\mathbb R^n$ is called polynomially integrable of degree $N$ if its parallel section functions are polynomials of degree $N$. We prove that the only smooth convex bodies with this property in odd dimensions are ellipsoids, if $N\ge n-1$. This is in contrast with the case of even dimensions and the case of odd dimensions with $NRead More

We consider the problem of finding the probability that a random triangle is obtuse, which was first raised by Lewis Caroll. Our investigation leads us to a natural correspondence between plane polygons and the Grassmann manifold of 2-planes in real $n$-space proposed by Allen Knutson and Jean-Claude Hausmann. This correspondence defines a natural probability measure on plane polygons. Read More

We prove that a sequence of quasi-Fuchsian representations for which the critical exponent converges to the topological dimension of the boundary of the group (larger than 2), converges up to subsequence and conjugacy to a totally geodesic representation. Read More

In \cite{gjha} we give a detailed analysis of spherical Hausdorff measures on sub-Riemannian manifolds in a general framework, that is, without the assumption of equiregularity. The present paper is devised as a complement of this analysis, with both new results and open questions. The first aim is to extend the study to other kinds of intrinsic measures on sub-Riemannian manifolds, namely Popp's measure and general (i. Read More

We survey problems and results from combinatorial geometry in normed spaces, concentrating on problems that involve distances. These include various properties of unit-distance graphs, minimum-distance graphs, diameter graphs, as well as minimum spanning trees and Steiner minimum trees. In particular, we discuss translative kissing (or Hadwiger) numbers, equilateral sets, and the Borsuk problem in normed spaces. Read More

For given convex integrands $\gamma_{{}_{i}}: S^{n}\to \mathbb{R}_{+}$ (where $i=1, 2$), the functions $\gamma_{{}_{max}}$ and $\gamma_{{}_{min}}$ can be defined as natural way. In this paper, we show that the Wulff shape of $\gamma_{{}_{max}}$ (resp. the Wulff shape of $\gamma_{{}_{min}}$) is exactly the convex hull of $(\mathcal{W}_{\gamma_{{}_{1}}}\cup \mathcal{W}_{\gamma_{{}_{2}}})$ (resp. Read More

We define a general framework that includes objects such as tilings, Delone sets, functions and measures, as a first step to discuss their almost periodicity in a systematic way. We define local derivability and mutual local derivability (MLD) between any two of these objects in order to describe their interrelation. We show several canonical maps in aperiodic order send an object P to a one that is MLD with P. Read More

The investigation of the relation among the distances of an arbitrary point in the Euclidean space $\mathbb{R}^n$ to the vertices of a regular $n$-simplex in that space has led us to the study of simplices having a regular facet. Calling an $n$-simplex with a regular facet an $n$-pre-kite, we investigate, in the spirit of [4], [10], [9], and [15], and using tools from linear algebra, the degree of regularity implied by the coincidence of any two of the classical centers of such simplices. We also prove that if $n \ge 3$, then the intersection of the family of $n$-pre-kites with any of the four known special families is the family of $n$-kites, thus extending the result in [18]. Read More

We show that there exist reduced polytopes in three-dimensional Euclidean space. This partially answers the question posed by Lassak on the existence of reduced polytopes in $d$-dimensional Euclidean space for $d\geq 3$. Moreover, we prove a novel necessary condition on reduced polytopes in three-dimensional Euclidean space. Read More

Let $(x_n)_{n=1}^{\infty}$ be a sequence on the torus $\mathbb{T}$ (normalized to length 1). We show that if there exists a sequence of positive real numbers $(t_n)_{n=1}^{\infty}$ converging to 0 such that $$ \lim_{N \rightarrow \infty}{\frac{1}{N^2} \sum_{m,n = 1}^{N}{\frac{1}{\sqrt{t_N}} \exp{\left(- \frac{1}{t_N} (x_m - x_n)^2 \right)}}} = \sqrt{\pi},$$ then $(x_n)_{n=1}^{\infty}$ is uniformly distributed. This is especially interesting when $t_N \sim N^{-2}$ since the size of the sum is then essentially determined exclusively by local gaps at scale $\sim N^{-1}$. Read More