Mathematics - Metric Geometry Publications (50)


Mathematics - Metric Geometry Publications

We study the intimate relationship between the Penrose and the Taylor-Socolar tilings, within both the context of double hexagon tiles and the algebraic context of hierarchical inverse sequences of triangular lattices. This unified approach produces both types of tilings together, clarifies their relationship, and offers straightforward proofs of their basic properties. Read More

We consider finite point subsets (distributions) in compact metric spaces. In the case of general rectifiable metric spaces, non-trivial bounds for sums of distances between points of distributions and for discrepancies of distributions in metric balls are given (Theorem 1.1). Read More

In this paper, we prove pointwise convergence of heat kernels for mGH-convergent sequences of $RCD^*(K,N)$-spaces. We obtain as a corollary results on the short-time behavior of the heat kernel in $RCD^*(K,N)$-spaces. We use then these results to initiate the study of Weyl's law in the $RCD$ setting Read More

The idea is to demonstrate the beauty and power of Alexandrov geometry by reaching interesting applications with a minimum of preparation. The applications include 1. Estimates on the number of collisions in billiards. Read More

We use a coarse version of the fundamental group first introduced by Barcelo, Capraro and White to show that box spaces of finitely presented groups detect the normal subgroups used to construct the box space, up to isomorphism. As a consequence we have that two finitely presented groups admit coarsely equivalent box spaces if and only if they are commensurable via normal subgroups. We also provide an example of two filtrations $(N_i)$ and $(M_i)$ of a free group $F$ such that $M_i>N_i$ for all $i$ with $[M_i:N_i]$ uniformly bounded, but with $\Box_{(N_i)}F$ not coarsely equivalent to $\Box_{(M_i)}F$. Read More

Let $T\subset{\mathbb R}^n$ be a fixed set. By a scaled copy of $T$ around $x\in{\mathbb R}^n$ we mean a set of the form $x+rT$ for some $r>0$. In this survey paper we study results about the following type of problems: How small can a set be if it contains a scaled copy of $T$ around every point of a set of given size? We will consider the cases when $T$ is circle or sphere centered at the origin, Cantor set in ${\mathbb R}$, the boundary of a square centered at the origin, or more generally the $k$-skeleton ($0\le kRead More

This paper introduces a notion of differential forms on closed, potentially fractal, subsets of Euclidean space by defining pointwise cotangent spaces using the restriction of $C^1$ functions to this set. Aspects of cohomology are developed: it is shown that the differential forms are a Banach algebra and it is possible to integrate these forms along rectifiable paths. These definitions are connected to the theory of differential forms on Dirichlet spaces by considering fractals with finite energy coordinates. Read More

In this paper, we will prove the Weyl's law for the asymptotic formula of Dirichlet eigenvalues on metric measure spaces with generalized Ricci curvature bounded from below. Read More

In this paper, we will establish an elliptic local Li-Yau gradient estimate for weak solutions of the heat equation on metric measure spaces with generalized Ricci curvature bounded from below. One of its main applications is a sharp gradient estimate for the logarithm of heat kernels. These results seem new even for smooth Riemannian manifolds. Read More

For a unimodular random graph $(G,\rho)$, we consider deformations of its intrinsic path metric by a (random) weighting of its vertices. This leads to the notion of the conformal growth exponent of $(G,\rho)$, which is the best asymptotic degree of volume growth of balls that can be achieved by such a reweighting. Under moment conditions on the degree of the root, we show that the conformal growth exponent of a unimodular random graph coincides with its almost sure spectral dimension (whenever the latter exists). Read More

It is a crucial problem in robotics field to cage an object using robots like multifingered hand. However the problem what is the caging for general geometrical objects and robots has not been well-described in mathematics though there were many rigorous studies on the methods how to cage an object by certain robots. In this article, we investigate the caging problem more mathematically and describe the problem in terms of recursion of the simple euclidean moves. Read More

Our aim is to find the minimal Hausdorff dimension of the union of scaled and/or rotated copies of the $k$-skeleton of a fixed polytope centered at the points of a given set. For many of these problems, we show that a typical arrangement in the sense of Baire category gives minimal Hausdorff dimension. In particular, this proves a conjecture of R. Read More

We consider the problem of comparing the volumes of two star bodies in an even-dimensional euclidean space $\mathbb R^{2n} = \mathbb C^n$ by comparing their cross sectional areas along complex lines (special 2-dimensional real planes) through the origin. Under mild symmetry conditions on one of the bodies a Busemann-Petty type theorem holds. Quaternionic and Octonionic analogs also hold. Read More

The subpower Higson corona of a proper metric space is defined in [J. Kucab, M. Zarichnyi, Subpower Higson corona of a metric space, Algebra and Discrete Mathematics 17(2014) n2, 280--287]. Read More

Let P be a polygon with rational vertices in the plane. We show that for any finite odd-sized collection of translates of P, the area of the set of points lying in an odd number of these translates is bounded away from 0 by a constant depending on P alone. The key ingredient of the proof is a construction of an odd cover of the plane by translates of P. Read More

The discrete Heisenberg group $\mathbb{H}_{\mathbb{Z}}^{2k+1}$ is the group generated by $a_1,b_1,\ldots,a_k,b_k,c$, subject to the relations $[a_1,b_1]=\ldots=[a_k,b_k]=c$ and $[a_i,a_j]=[b_i,b_j]=[a_i,b_j]=[a_i,c]=[b_i,c]=1$ for every distinct $i,j\in \{1,\ldots,k\}$. Denote $S=\{a_1^{\pm 1},b_1^{\pm 1},\ldots,a_k^{\pm 1},b_k^{\pm 1}\}$. The horizontal boundary of $\Omega\subset \mathbb{H}_{\mathbb{Z}}^{2k+1}$, denoted $\partial_{h}\Omega$, is the set of all $(x,y)\in \Omega\times (\mathbb{H}_{\mathbb{Z}}^{2k+1}\setminus \Omega)$ such that $x^{-1}y\in S$. Read More

By a geodesic subspace of a metric space $X$ we mean a subset $A$ of $X$ such that any two points in $A$ can be connected by a geodesic in $A$. It is easy to check that a geodesic metric space $X$ is an $\mathbb{R}$-tree (that is, a $0$-hyperbolic space in the sense of Gromov) if and only if the union of any two intersecting geodesic subspaces is again a geodesic subspace. In this paper, we prove an analogous characterization of general Gromov hyperbolic spaces, where we replace geodesic subspaces by quasigeodesic subspaces. Read More

We study Morse representations of discrete subgroups in higher rank semi-simple Lie groups defined by M. Kapovich, B. Leeb and J. Read More

In this article we introduce and study uniform and non-uniform approximate lattices in locally compact second countable (lcsc) groups. These are approximate subgroups (in the sense of Tao) which simultaneously generalize lattices in lcsc group and mathematical quasi-crystals (a.k. Read More

The tensorial curvature measures are tensor-valued generalizations of the curvature measures of convex bodies. In a previous work, we obtained kinematic formulae for all tensorial curvature measures. As a consequence of these results, we now derive a complete system of Crofton formulae for such tensorial curvature measures on convex bodies and for their (nonsmooth) generalizations on convex polytopes. Read More

This is a chapter of a forthcoming Lecture Notes in Mathematics "Modern Approaches to Discrete Curvature" edited by L. Najman and P. Romon. Read More

Tensorial curvature measures are tensor-valued generalizations of the curvature measures of convex bodies. We prove a complete set of kinematic formulae for such tensorial curvature measures on convex bodies and for their (nonsmooth) generalizations on convex polytopes. These formulae express the integral mean of the tensorial curvature measure of the intersection of two given convex bodies (resp. Read More

In this paper, we give a simple criterion for a locally distributive semilattice to have CAT(0) orthoscheme complex. Namely, the orthoscheme complex of a locally distributive semilattice S is CAT(0) if and only if S is a flag semilattice, that is, any pairwise bounded triple of S is bounded. Read More

For a nonempty polyhedral set $P\subset \mathbb R^d$, let $\mathcal F(P)$ denote the set of faces of $P$, and let $N(P,F)$ be the normal cone of $P$ at the nonempty face $F\in\mathcal F(P)$. We prove that the function $\sum_{F\in\mathcal F(P)}(-1)^{\text{dim} F} 1_{F-N(P,F)}$ equals $1$ if $P$ is bounded, or $0$ if $P$ is unbounded and line-free. Previously, this formula was known to hold everywhere outside some exceptional set of Lebesgue measure $0$ or for polyhedral cones. Read More

The Lott-Sturm-Villani Curvature-Dimension condition provides a synthetic notion for a metric-measure space to have Ricci-curvature bounded from below and dimension bounded from above. We prove that it is enough to verify this condition locally: an essentially non-branching metric-measure space $(X,{\mathsf d},{\mathfrak m})$ (so that $({\mathfrak m},{\mathsf d})$ is a length-space and ${\mathfrak m}(X) < \infty$) verifying the local Curvature-Dimension condition $\mathsf{CD}_{loc}(K,N)$ with parameters $K \in \mathbb{R}$ and $N \in (1,\infty)$, also verifies the global Curvature-Dimension condition $\mathsf{CD}(K,N)$, meaning that the Curvature-Dimension condition enjoys the globalization (or local-to-global) property. The main new ingredients of our proof are an explicit $\textit{change-of-variables}$ formula for densities of Wasserstein geodesics depending on a second-order derivative of an associated Kantorovich potential; a surprising $\textit{third-order}$ bound on the latter Kantorovich potential, which holds in complete generality on any proper geodesic space; and a certain $\textit{rigidity}$ property of the change-of-variables formula, allowing us to bootstrap the a-priori available regularity. Read More

For a finite set $X$ of $n$ points from $\mathbb{R}^M$, the degree of an $M$-element subset $\{x_1,\dots,x_M\}$ of $X$ is defined as the number of $M$-simplices that can be constructed from this $M$-element subset using an additional point $z\in X$, such that no further point of $X$ lies in the interior of this $M$-simplex. Furthermore, the degree of $X$, denoted by $\textrm{deg} (X)$, is the maximal degree of any of its $M$-element subsets. The purpose of this paper is to show that the moments of the degree of $X$ satisfy $\mathbb{E}\left[\textrm{deg} (X)^k\right] \geq c n^k / \log n$, for some constant $c>0$, if the elements of the set $X$ are chosen uniformly from a convex body $W \subset \mathbb{R}^M$. Read More

We prove several new results around Gromov's waist theorem. We consider waists of different Riemannian manifolds, real and complex projective spaces, flat tori, convex bodies in the Euclidean space. We also make an effort to establish certain waist-type results in terms of the Hausdorff measure. Read More

We introduce matrix algebra of subsets in metric spaces and we apply it to improve results of Yamauchi and Davila regarding Asymptotic Property C. Here is a representative result: Suppose $X$ is an $\infty$-pseudo-metric space and $n\ge 0$ is an integer. The asymptotic dimension of $X$ is at most $n$ if and only if for any real number $r > 0$ and any integer $m\ge 1$ there is an augmented $m\times (n+1)$-matrix $\mathcal{M}=[\mathcal{B} |\mathcal{A}]$ (that means $\mathcal{B}$ is a column-matrix and $\mathcal{A}$ is an $m\times n$-matrix) of subspaces of $X$ of scale-$r$-dimension $0$ such that $\mathcal{M}\cdot_\cap \mathcal{M}^T$ is bigger than or equal to the identity matrix and $B(\mathcal{A},r)\cdot_\cap B(\mathcal{A},r)^T$ is a diagonal matrix. Read More

This paper considers the radii functionals (circumradius, inradius, and diameter) as well as the Minkowski asymmetry for general (possibly non-symmetric) gauge bodies. A generalization of the concentricity inequality (which states that the sum of the inradius and circumradius is not greater than the diameter in general Minkowski spaces) for non-symmetric gauge bodies is derived and a strong connection between this new inequality, extremal sets of the generalized Bohnenblust inequality, and completeness of simplices is revealed. Read More

We define a class of trim metric spaces and show that every finite metric space is the leaf space of a metric forest with trim base. Read More

We give a characterization for asymptotic dimension growth. We apply it to CAT(0) cube complexes of finite dimension, giving an alternative proof of N. Wright's result on their finite asymptotic dimension. Read More

We prove that flow of a generic geodesic on a flat surface with finite holonomy group is ergodic. We use this result to prove that flows of generic billiards on certain flat surfaces with boundary are also ergodic. Read More

In the setting of a metric space $X$ equipped with a doubling measure that supports a Poincar\'e inequality, we show that if $u_i\to u$ strictly in $BV(X)$, i.e. if $u_i\to u$ in $L^1(X)$ and $\Vert Du_i\Vert(X)\to\Vert Du\Vert(X)$, then for a subsequence (not relabeled) we have $\widetilde{u}_i(x)\to \widetilde{u}(x)$ for $\mathcal H$-almost every $x\in X\setminus S_u$. Read More

In the setting of a metric space equipped with a doubling measure that supports a Poincar\'e inequality, we show that a set $E$ is of finite perimeter if and only if $\mathcal H(\partial^1 I_E)<\infty$, that is, if and only if the codimension one Hausdorff measure of the \emph{$1$-fine boundary} of the set's measure theoretic interior $I_E$ is finite. Read More

Let $P_{n,2d}$ be the vector space of homogenous real polynomials with $n$ variables and degree $2d$. Let $ P_{A} \subset P_{n,2d}$ be the subspace spanned by polynomials $A=\{ r_1,r_2, \ldots, r_{m} \}$, where we only assume $r_1=(x_1^2+x_2^2+\ldots+x_n^2)^{d}$. The elements of $P_{A}$ that are nonnegative on the sphere, forms a full dimensional cone that we denote by $Pos_{A}$. Read More

Every set of points $\mathcal{P}$ determines $\Omega(|\mathcal{P}| / \log |\mathcal{P}|)$ distances. A close version of this was initially conjectured by Erd\H{o}s in 1946 and rather recently proved by Guth and Katz. We show that when near this lower bound, a point set $\mathcal{P}$ of the form $A \times A$ must satisfy $|A - A| \ll |A|^{2-\frac{2}{7}} \log^{\frac{1}{7}} |A|$. Read More

We study two notions of Dirichlet problem associated with BV energy minimizers (also called functions of least gradient) in bounded domains in metric measure spaces whose measure is doubling and supports a $(1,1)$-Poincar\'e inequality. Since one of the two notions is not amenable to the direct method of the calculus of variations, we construct, based on an approach of [23, 29], solutions by considering the Dirichlet problem for $p$-harmonic functions, $p>1$, and letting $p\to 1$. Tools developed and used in this paper include the inner perimeter measure of a domain. Read More

This short note describes a connection between algorithmic dimensions of individual points and classical pointwise dimensions of measures. Read More

We give a completely constructive solution to Tarski's circle squaring problem. More generally, we prove a Borel version of an equidecomposition theorem due to Laczkovich. If $k \geq 1$ and $A, B \subseteq \mathbb{R}^k$ are bounded Borel sets with the same positive Lebesgue measure whose boundaries have upper Minkowski dimension less than $k$, then $A$ and $B$ are equidecomposable by translations using Borel pieces. Read More

We consider the number of distinct distances between two finite sets of points in ${\bf R}^k$, for any constant dimension $k\ge 2$, where one set $P_1$ consists of $n$ points on a line $l$, and the other set $P_2$ consists of $m$ arbitrary points, such that no hyperplane orthogonal to $l$ and no hypercylinder having $l$ as its axis contains more than $O(1)$ points of $P_2$. The number of distinct distances between $P_1$ and $P_2$ is then $$ \Omega\left(\min\left\{ n^{2/3}m^{2/3},\; \frac{n^{10/11}m^{4/11}}{\log^{2/11}m},\; n^2,\; m^2\right\}\right) . $$ Without the assumption on $P_2$, there exist sets $P_1$, $P_2$ as above, with only $O(m+n)$ distinct distances between them. Read More

In this paper we provide an extension of a theorem of David and Semmes ('91) to general non-atomic measures. The result provides a geometric characterization of the non-atomic measures for which a certain class of square function operators, or singular integral operators, are bounded in $L^2(\mu)$. The description is given in terms of a modification of Jones' $\beta$-coefficients. Read More

This paper presents bounds for the best approximation, with respect to the Hausdorff metric, of a convex body $K$ by a circumscribed polytope $P$ with a given number of facets. These bounds are of particular interest if $K$ is elongated. To measure the elongation of the convex set, its isoperimetric ratio $ V_j(K)^{1/j} V_i(K)^{-1/i} $ is used. Read More

A pose of a rigid object is usually regarded as a rigid transformation, described by a translation and a rotation. In this article, we define a pose as a distinguishable static state of the considered object, and show that the usual identification of the pose space with the space of rigid transformations is abusive, as it is not adapted to objects with proper symmetries. Based solely on geometric considerations, we propose a frame-invariant metric on the pose space, valid for any physical object, and requiring no arbitrary tuning. Read More

In $n$-dimensional hyperbolic space $\mathbf{H}^n$ $(n\ge2)$ there are $3$-types of spheres (balls): the sphere, horosphere and hypersphere. If $n=2,3$ we know an universal upper bound of the ball packing densities, where each ball volume is related to the volume of the corresponding Dirichlet-Voronoi (D-V) cell. E. Read More

It is shown that self avoiding walk on the seven regular infinite planar triangulation has linear expected displacement. Read More

In this paper we consider the root set of power series with finite choices of coefficients: $\{z\in\mathbb{C}| \exists a_n\in H, \sum_{n=0}^{\infty} a_nz^n=0\}$ where $H\subset\mathbb{C}$ is a finite subset. For $H=\{\pm 1\}$, the root set of Littlewood series is under consideration for a long while. Here we study the case when $H=\{e^{i\frac{2\pi}{p}k},k=0,1,2,. Read More

We prove the multivariate Fujiwara bound for exponential sums: for a $d$-variate exponential sum $f$ with scaling parameter $\mu$, if $x$ is contained in the amoeba $\mathscr{A}(f)$, then the distance from $x$ to the Archimedean tropical variety associated to $f$ is at most $d \sqrt{d}\, 2\log(2 + \sqrt{3})/ \mu$. If $f$ is polynomial, then the bound can be improved to $d \log(2 + \sqrt{3})$. Read More

This paper proposes a general study of the asymptotic geometry of groups with contracting elements, with the emphasis on a subclass of \textit{statistically convex-cocompact} action. We prove that statistically convex-cocompact actions have the following properties: A. They contains purely contracting, quasi-convex, large free sub-semigroups, B. Read More