Mathematics - Metric Geometry Publications (50)

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Mathematics - Metric Geometry Publications

Denote by ${\mathcal K}^d$ the family of convex bodies in $E^d$ and by $w(C)$ the minimal width of $C \in {\mathcal K}^d$. We ask for the greatest number $\Lambda_n ({\mathcal K}^d)$ such that every $C \in {\mathcal K}^d$ contains a polytope $P$ with at most $n$ vertices for which $\Lambda_n ({\mathcal K}^d) \leq \frac{w(P)}{w(C)}$. We give a lower estimate of $\Lambda_n ({\mathcal K}^d)$ for $n \geq 2d$ based on estimates of the smallest radius of $\big\lfloor {\frac{n}{2}} \big\rfloor$ antipodal pairs of spherical caps that cover the unit sphere of $E^d$. Read More


In this paper we establish a general form of the Mass Transference Principle for systems of linear forms conjectured in [1]. We also present a number of applications to problems in Diophantine approximation. These include a general transference of Khintchine-Groshev type theorems into Hausdorff measure statements. Read More


The present article addresses to everyone who starts working with (pointed) Gromov-Hausdorff convergence. In the major part, both Gromov-Hausdorff convergence of compact and of pointed metric spaces are introduced and investigated. Moreover, the relation of sublimits occurring with pointed Gromov-Hausdorff convergence and ultralimits is discussed. Read More


WDC sets in ${\mathbb R}^d$ were recently defined as sublevel sets of DC functions (differences of convex functions) at weakly regular values. They form a natural and substantial generalization of sets with positive reach and still admit the definition of curvature measures. Using results on singularities of convex functions, we obtain regularity results on the boundaries of WDC sets. Read More


We generalize Cauchy's celebrated theorem on the global rigidity of convex polyhedra in Euclidean $3$-space $\mathbb{E}^{3}$ to the context of circle polyhedra in the $2$-sphere $\mathbb{S}^{2}$. We prove that any two convex and bounded non-unitary \textit{c}-polyhedra with M\"obius-congruent faces that are consistently oriented are M\"obius-congruent. Our result implies the global rigidity of convex inversive distance circle packings as well as that of certain hyperideal hyperbolic polyhedra. Read More


We revisit the problem of characterizing the eigenvalue distribution of the Dirichlet-Laplacian on bounded open sets $\Omega\subset\mathbb{R}$ with fractal boundaries. It is well-known from the results of Lapidus and Pomerance \cite{LapPo1} that the asymptotic second term of the eigenvalue counting function can be described in terms of the Minkowski content of the boundary of $\Omega$ provided it exists. He and Lapidus \cite{HeLap2} discussed a remarkable extension of this characterization to sets $\Omega$ with boundaries that are not necessarily Minkowski measurable. Read More


In this article we investigate the pressure function and affinity dimension for iterated function systems associated to the "box-like" self-affine fractals investigated by D.-J. Feng, Y. Read More


Schmidt's game is generally used to to deduce qualitative information about the Hausdorff dimensions of fractal sets and their intersections. However, one can also ask about quantitative versions of the properties of winning sets. In this paper we show that such quantitative information has applications to various questions including: * What is the maximal length of an arithmetic progression on the "middle $\epsilon$" Cantor set? * What is the smallest $n$ such that there is some element of the ternary Cantor set whose continued fraction partial quotients are all $\leq n$? * What is the Hausdorff dimension of the set of $\epsilon$-badly approximable numbers on the Cantor set? We show that a variant of Schmidt's game known as the $potential$ $game$ is capable of providing better bounds on the answers to these questions than the classical Schmidt's game. Read More


We prove that for $n\geq 2$, the Lipschitz homotopy group $\pi_{n+1}^{\rm lip}(\mathbb{H}^n)\neq 0$ of the Heisenberg group $\mathbb{H}^n$ is nontrivial. Read More


The notion of a valuation on convex bodies is very classical. The notion of a valuation on a class of functions was recently introduced and studied by M. Ludwig and others. Read More


A lattice in the Euclidean space is standard if it has a basis consisting vectors whose norms equal to the length in its successive minima. In this paper, it is shown that with the $L^2$ norm all lattices of dimension $n$ are standard if and only if $n\leqslant 4$. It is also proved that with an arbitrary norm, every lattice of dimensions 1 and 2 is standard. Read More


Contact graphs have emerged as an important tool in the study of translative packings of convex bodies. The contact number of a packing of translates of a convex body is the number of edges in the contact graph of the packing, while the Hadwiger number of a convex body is the maximum vertex degree over all such contact graphs. In this paper, we investigate the Hadwiger and contact numbers of totally separable packings of convex domains, which we refer to as the separable Hadwiger number and the separable contact number, respectively. Read More


We establish a refinement of Marstrand's projection theorem for Hausdorff dimension functions finer than the usual power functions, including an analogue of Marstrand's Theorem for logarithmic Hausdorff dimension. Read More


We attach buildings to modular lattices and use them to develop a metric approach to Harder-Narasimhan filtrations. Switching back to a categorical framework, we establish an abstract numerical criterion for the compatibility of these filtrations with tensor products. We finally verify our criterion in three cases, one of which is new. Read More


We show that the pointed measured Gromov convergence of the underlying spaces implies (or under some condition, is equivalent to) the weak convergence of Brownian motions under Riemannian Curvature-Dimension (RCD) conditions. This paper is an improved and jointed version of the previous two manuscripts arXiv:1509.02025 and arXiv:1603. Read More


In this paper, the dual Orlicz curvature measure is proposed and its basic properties are provided. A variational formula for the dual Orlicz-quermassintegral is established in order to give a geometric interpretation of the dual Orlicz curvature measure. Based on the established variational formula, a solution to the dual Orlicz-Minkowski problem regarding the dual Orlicz curvature measure is provided. Read More


We show that the set of elusive singularities attached to the T-fractal surface by its metric completion forms a Cantor set, and are wild singularities in the sense of Bowman and Valdez. We compute the Hausdorff dimension of this set of singularities, and show each of these singularities has infinitely-many rotational components. Read More


We consider the Assouad dimension analogues of two important problems in geometric measure theory. These problems are tied together by the common theme of `passing to weak tangents'. First, we solve the analogue of Falconer's distance set problem for Assouad dimension in the plane: if a planar set has Assouad dimension greater than 1, then its distance set has Assouad dimension 1. Read More


We study real-valued, continuous and translation invariant valuations defined on the space of quasi-concave functions of N variables. In particular, we prove a homogeneous decomposition theorem of McMullen type, and we find a representation formula for those valuations which are N-homogeneous. Moreover, we introduce the notion of Klain's functions for these type of valuations. Read More


A one-to-one correspondence between the infinitesimal motions of bar-joint frameworks in $\mathbb{R}^d$ and those in $\mathbb{S}^d$ is a classical observation by Pogorelov, and further connections among different rigidity models in various different spaces have been extensively studied. In this paper, we shall extend this line of research to include the infinitesimal rigidity of frameworks consisting of points and hyperplanes. This enables us to understand correspondences between point-hyperplane rigidity, classical bar-joint rigidity, and scene analysis. Read More


After having investigated the geodesic and translation triangles and their angle sums in $\NIL$ and $\SLR$ geometries we consider the analogous problem in $\SOL$ space that is one of the eight 3-dimensional Thurston geometries. We analyse the interior angle sums of translation triangles in $\SOL$ geometry and prove that it can be larger or equal than $\pi$. In our work we will use the projective model of $\SOL$ described by E. Read More


All continuous, SL$(n)$ and translation invariant valuations on the space of convex functions on ${\mathbb R}^n$ are completely classified. Read More


We prove that if $\Phi:X\to Y$ a mapping of weak bounded length distortion from a quasiconvex and complete metric space $X$ to any metric space $Y$, then for any Lipschitz mapping $f:\mathbb{R}^k\supset E\to X$ we have that ${\mathcal H}^k(f(E))=0$ in $X$ if and only if ${\mathcal H}^k(\Phi(f(E)))=0$ in $Y$. This generalizes an earlier result of Haj\l{}asz and Malekzadeh where the target space $Y$ was a Euclidean space $Y=\mathbb{R}^N$. Read More


In this short note we improve the best to date bound in Godbersen's conjecture, and show some implications for unbalanced difference bodies. Read More


The dual Minkowski problem for even data asks what are the necessary and sufficient conditions on an even prescribed measure on the unit sphere for it to be the $q$-th dual curvature measure of an origin-symmetric convex body in $\mathbb{R}^n$. A full solution to this is given when $1 < q < n$. The necessary and sufficient condition is an explicit measure concentration condition. Read More


There were obtained some properties of weakly m-convex sets. Various kinds of the problem of shadow were investigated. There was obtained the lower estimation for the number of balls that are necessary to create a shadow at the point of the sphere in three-dimensional Euclidean space Read More


We verify a conjecture of Rajala: if $(X,d)$ is a metric surface of locally finite Hausdorff 2-measure admitting some (geometrically) quasiconformal parametrization by a simply connected domain $\Omega \subset \mathbb{R}^2$, then there exists a quasiconformal mapping $f: X \rightarrow \Omega$ satisfying the modulus inequality $2\pi^{-1}\text{Mod }\Gamma \leq \text{Mod }f\Gamma \leq 4\pi^{-1}\text{Mod }\Gamma$ for all curve families $\Gamma$ in $X$. This inequality is the best possible. Our proof is based on an inequality for the area of a planar convex body under a linear transformation which attains its Banach-Mazur distance to the Euclidean unit ball. Read More


Polyhedral surfaces are fundamental objects in architectural geometry and industrial design. Whereas closeness of a given mesh to a smooth reference surface and its suitability for numerical simulations were already studied extensively, the aim of our work is to find and to discuss suitable assessments of smoothness of polyhedral surfaces that only take the geometry of the polyhedral surface itself into account. Motivated by analogies to classical differential geometry, we propose a theory of smoothness of polyhedral surfaces including suitable notions of normal vectors, tangent planes, asymptotic directions, and parabolic curves that are invariant under projective transformations. Read More


In this note we give necessary and sufficient conditions for the validity of the local spectral convergence, in balls, on the $RCD^*$-setting. Read More


In the setting of a metric space that is equipped with a doubling measure and supports a Poincar\'e inequality, we show that the total variation of functions of bounded variation is lower semicontinuous with respect to $L^1$-convergence in every $1$-quasiopen set. To achieve this, we first prove a new characterization of the total variation in $1$-quasiopen sets. Then we utilize the lower semicontinuity to show that the variation measures of a sequence of functions of bounded variation converging in the strict sense are uniformly absolutely continuous with respect to the $1$-capacity. Read More


It is shown that a classical construction of D.S. Asche of 72 equiangular lines in the 19 dimensional Euclidean space contains a subset of 54 equiangular lines embedded in an 18 dimensional subspace. Read More


We prove a sub-Riemannian version of Bonnet-Myers theorem that applies to any quaternionic contact manifold of dimension bigger than 7. The curvature condition is expressed in terms of bounds on the horizontal component of standard tensors of quaternionic contact geometry. It is obtained by explicit computation of the sub-Riemannian curvature on quaternionic contact manifolds, defined as coefficients of the sub-Riemannian Jacobi equation. Read More


We prove that the total curvature of a planar graph with nonnegative combinatorial curvature is at least $\frac{1}{12}$ if it is positive, and show that this estimate is sharp. This answers a question proposed by T. R{\'e}ti. Read More


This paper addresses the problem of constructing bearing rigid networks in arbitrary dimensions. We show that the bearing rigidity of a network is a generic property that is critically determined by the underlying graph of the network. We define a new notion termed generic bearing rigidity for graphs. Read More


We construct the first examples of residually finite non-exact groups. The construction is based on author's earlier construction of groups containing isometrically expanders using a graphical small cancellation. Read More


In this dissertation we define a generalization of Kakeya sets in certain metric spaces. Kakeya sets in Euclidean spaces are sets of zero Lebesgue measure containing a segment of length one in every direction. A famous conjecture, known as Kakeya conjecture, states that the Hausdorff dimension of any Kakeya set should equal the dimension of the space. Read More


Let n points be placed on a closed convex domain on the plane, no three points on a straight line. Read More


We prove that the Teichm\"uller space of surfaces with given boundary lengths equipped with the arc metric (resp. the Teichm\"uller metric) is almost isometric to the Teichm\"uller space of punctured surfaces equipped with the Thurston metric (resp. the Teichm\"uller metric). Read More


Hamenst\"adt gave a parametrization of the Teichm\"uller space of punctured surfaces such that the image under this parametrization is the interior of a polytope. In this paper, we study the Hilbert metric on the Teichm\"uller space of punctured surfaces based on this parametrization. We prove that every earthquake ray is an almost geodesic under the Hilbert metric. Read More


We prove that finite perimeter subsets of $\mathbb{R}^{n+1}$ with small isoperimetric deficit have boundary Hausdorff-close to a sphere up to a subset of small measure. We also refine this closeness under some additional a priori integral curvature bounds. As an application, we answer a question raised by B. Read More


Let $X_1,\ldots,X_n$ be independent random points that are distributed according to a probability measure on $\mathbb{R}^d$ and let $P_n$ be the random convex hull generated by $X_1,\ldots,X_n$ ($n\geq d+1$). Natural classes of probability distributions are characterized for which, by means of Blaschke-Petkantschin formulae from integral geometry, one can show that the mean facet number of $P_n$ is strictly monotonically increasing in $n$. Read More


This presentation starts with the regular polygons, of course, then with the Platonic and Archimedean solids. The latter ones are whose symmetry groups are transitive on the vertices, and in addition, whose faces are regular polygons (see only I. Prok's home page [11] for them). Read More


Recall that a convex body $K$ is in John's position if the unit Euclidean ball is the maximal volume ellipsoid contained in $K$. Approximating convex body in John's position by polytopes we obtain the following results. 1. Read More


Let $1<\beta<2$. Given any $x\in[0, (\beta-1)^{-1}]$, a sequence $(a_n)\in\{0,1\}^{\mathbb{N}}$ is called a $\beta$-expansion of $x$ if $x=\sum_{n=1}^{\infty}a_n\beta^{-n}.$ For any $k\geq 1$ and any $(b_1b_2\cdots b_k)\in\{0,1\}^{k}$, if there exists some $k_0$ such that $a_{k_0+1}a_{k_0+2}\cdots a_{k_0+k}=b_1b_2\cdots b_k$, then we call $(a_n)$ a universal $\beta$-expansion of $x$. Read More


This survey is based on a series of lectures that we gave at MSRI in Spring 2015 and on a series of papers, mostly written jointly with Joan Porti. Our goal here is to: 1. Describe a class of discrete subgroups $\GammaRead More


Let $(X,d,\mu)$ be a doubling metric measure space endowed with a Dirichlet form $\mathscr{E}$ deriving from a "carr\'e du champ". Assume that $(X,d,\mu,\mathscr{E})$ supports a scale-invariant $L^2$-Poincar\'e inequality. In this article, we study the following properties of harmonic functions, heat kernels and Riesz transform for $p\in (2,\infty]$: (i) $(G_p)$: $L^p$-boundedness of the gradient of the associated heat semigroup; (ii) $(RH_p)$: $L^p$-reverse H\"older inequality for the gradient of harmonic functions; (iii) $(R_p)$: $L^p$-boundedness of the Riesz transform ($p<\infty$); (iv) $(GBE)$: a generalized Bakry-\'Emery condition. Read More


Motivated by Perelman's Pseudo Locality Theorem for the Ricci flow, we prove that if a Riemannian manifold has Ricci curvature bounded below in a metric ball which moreover has almost maximal volume, then in a smaller ball (in a quantified sense) it holds an almost-euclidean isoperimetric inequality. The result is actually established in the more general framework of non-smooth spaces satisfying local Ricci curvature lower bounds in a synthetic sense via optimal transportation. Read More


We study the geometry of hyperbolic cone surfaces, possibly with cusps or geodesic boundaries. We prove that any hyperbolic cone structure on a surface of non-exceptional type is determined up to isotopy by the geodesic lengths of a finite specific homotopy classes of non-peripheral simple closed curves. As an application, we show that the Thurston asymmetric metric is well-defined on the Teichm\"uller space of hyperbolic cone surfaces with fixed cone angles and boundary lengths. Read More


We consider the Carnot-Carath\'eodory distance $\delta_E$ to a closed set $E$ in the sub-Riemannian Heisenberg groups $\mathbb{H}^n$, $n\ge 1$. The $\mathbb{H}$-regularity of $\delta_E$ is proved under mild conditions involving a general notion of singular points. In case $E$ is a Euclidean $C^k$ submanifold, $k\ge 2$, we prove that $\delta_E$ is $C^k$ out of the singular set. Read More


The 'contracting boundary' of a proper geodesic metric space consists of equivalence classes of geodesic rays that behave like rays in a hyperbolic space. We introduce a geometrically relevant, quasi-isometry invariant topology on the contracting boundary. When the space is the Cayley graph of a finitely generated group we show that our new topology is metrizable. Read More