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Combinatorial properties of ultrametrics and generalized ultrametricsAug 22 2019Let $X$, $Y$ be sets and let $\Phi$, $\Psi$ be mappings with domains $X^{2}$ and $Y^{2}$ respectively. We say that $\Phi$ and $\Psi$ are combinatorially similar if there are bijections $f \colon \Phi(X^2) \to \Psi(Y^{2})$ and $g \colon Y \to X$ such that ... More
On Spaces of Inscribed TrianglesAug 22 2019Meyerson's Theorem says that all but at most 2 points of any Jordan loop are vertices of inscribed equilateral triangles. We show that for any Jordan loop there are uncountable many other triangle shapes for which this same result is true. Our result ... More
Tropical Ehrhart Theory and Tropical VolumeAug 21 2019We introduce a novel intrinsic volume concept in tropical geometry. This is achieved by developing the foundations of a tropical analog of lattice point counting in polytopes. We exhibit the basic properties and compare it to existing measures. Our exposition ... More
A $C^m$ Lusin Approximation Theorem for Horizontal Curves in the Heisenberg GroupAug 20 2019We prove a $C^m$ Lusin approximation theorem for horizontal curves in the Heisenberg group. This states that every absolutely continuous horizontal curve whose horizontal velocity is $m-1$ times $L^1$ differentiable almost everywhere coincides with a ... More
The doubling metric and doubling measuresAug 20 2019We introduce the so--called doubling metric on the collection of non--empty bounded open subsets of a metric space. Given a subset $U$ of a metric space $X$, the predecessor $U_{*}$ of $U$ is defined by doubling the radii of all open balls contained inside ... More
On the structure of RCD spaces with upper curvature boundsAug 19 2019We develop a structure theory for RCD spaces with curvature bounded above in Alexandrov sense. In particular, we show that any such space is a topological manifold with boundary whose interior is equal to the set of regular points. Further the set of ... More
On Orlicz-Jensen-Hermite-Hadamard inequalities and their applications in Convex GeometryAug 18 2019In this paper we show some Orlicz-Jensen-Hermite-Hadamard inequality and a reverse to that inequality. This establishes, in particular, one of the first multidimensional Hermite-Hadamard inequality in this generality. We then show several consequences ... More
Poincaré Inequalities and Uniform RectifiabilityAug 18 2019We show that any $d$-Ahlfors regular subset of $\mathbb{R}^{n}$ supporting a weak $(1,d)$-Poincar\'e inequality with respect to surface measure is uniformly rectifiable.
The Heintze-Karcher inequality for metric measure spacesAug 16 2019In this note we prove the Heintze-Karcher inequality for essentially non-branching metric measure spaces satisfying a lower Ricci curvature bound in the sense of Lott-Sturm-Villani. The proof is based on the the needle decomposition technique for metric ... More
On the union of essentially distinct $δ$-tubesAug 16 2019We say two $\delta$-tubes (dimension $\delta\times\cdots\times\delta\times1$) in $\mathbb{R}^n$ are essentially distinct if the measure of their intersection is smaller than a half of a single $\delta$-tube. For a collection of essentially distinct $\delta$-tubes, ... More
Laplacian algebras, manifold submetries and the Inverse Invariant Theory ProblemAug 15 2019Manifold submetries of the round sphere are a class of partitions of the round sphere that generalizes both singular Riemannian foliations, and the orbit decompositions by the orthogonal representations of compact groups. We exhibit a one-to-one correspondence ... More
The maximum number of points in the cross-polytope that form a packing set of a scaled cross-polytopeAug 15 2019The problem of finding the largest number of points in the unit cross-polytope such that the $l_{1}$-distance between any two distinct points is at least $2r$ is investigated for $r\in\left(1-\frac{1}{n},1\right]$ in dimensions $\geq2$ and for $r\in\left(\frac{1}{2},1\right]$ ... More
The conic geometry of rectangles inscribed in linesAug 15 2019We develop a circle of ideas involving pairs of lines in the plane, intersections of hyperbolically rotated elliptical cones and the locus of the centers of rectangles inscribed in lines in the plane.
Envelope PolyhedraAug 15 2019This paper presents an additional class of regular polyhedra--envelope polyhedra--made of regular polygons, where the arrangement of polygons (creating a single surface) around each vertex is identical; but dihedral angles between faces need not be identical, ... More
Quadratic Split Quaternion Polynomials: Factorization and GeometryAug 14 2019We investigate factorizability of a quadratic split quaternion polynomial. In addition to inequality conditions for existence of such factorization, we provide lucid geometric interpretations in the projective space over the split quaternions.
Cosmology of Plane GeometryAug 14 2019This paper focuses on a new approach to plane geometry and develops important concepts that can allow researchers to unite and observe plane geometry from a new, meaningful perspective.
Improved bounds for the Kakeya maximal conjecture in higher dimensionsAug 14 2019We adapt Guth's polynomial partitioning argument for the Fourier restriction problem to the context of the Kakeya problem. By writing out the induction argument as a recursive algorithm, additional multiscale geometric information is made available. To ... More
A Gordian Pair of LinksAug 13 2019We construct a pair of isotopic link configurations that are not thick isotopic while preserving total length.
$p$-adic Integral GeometryAug 13 2019We prove a $p$-adic version of the Integral Geometry Formula for averaging the intersection of two $p$-adic projective algebraic sets. We apply this result to give bounds on the number of points in the modulo $p^m$ reduction of a projective set (reproving ... More
Bounded geometry and $p$-harmonic functions under uniformization and hyperbolizationAug 13 2019The uniformization and hyperbolization transformations formulated by Bonk, Heinonen and Koskela in \emph{"Uniformizing Gromov Hyperbolic Spaces"}, Ast\'erisque {\bf 270} (2001), dealt with geometric properties of metric spaces. In this paper we consider ... More
The Quasi-hyperbolicity Constant of a Metric SpaceAug 12 2019We introduce the quasi-hyperbolicity constant of a metric space, a rough isometry invariant that measures how a metric space deviates from being Gromov hyperbolic. This number, for unbounded spaces, lies in the closed interval $[1,2]$. The quasi-hyperbolicity ... More
Quantitative combinatorial geometry for concave functionsAug 12 2019We prove several exact quantitative versions of Helly's and Tverberg's theorems, which guarantee that a finite family of convex sets in $R^d$ has a large intersection. Our results characterize conditions that are sufficient for the intersection of a family ... More
On the Lipschitz dimension of Cheeger-KleinerAug 12 2019In a 2013 paper, Cheeger and Kleiner introduced a new type of dimension for metric spaces, the "Lipschitz dimension". We study the dimension-theoretic properties of Lipschitz dimension, including its behavior under Gromov-Hausdorff convergence, its (non-)invariance ... More
Facets of spherical random polytopesAug 12 2019Facets of the convex hull of $n$ independent random vectors chosen uniformly at random from the unit sphere in $\mathbb{R}^d$ are studied. A particular focus is given on the height of the facets as well as the expected number of facets as the dimension ... More
Almost-rigidity of frameworksAug 10 2019We extend the mathematical theory of rigidity of frameworks (graphs embedded in $d$-dimensional space) to consider nonlocal rigidity and flexibility properties. We provide conditions on a framework under which (I) as the framework flexes continuously ... More
Abstract FractalsAug 09 2019We develop a new definition of fractals which can be considered as an abstraction of the fractals determined through self-similarity. The definition is formulated through imposing conditions which are governed the relation between the subsets of a metric ... More
Convergence of symmetrization processesAug 08 2019Steiner and Schwarz symmetrizations, and their most important relatives, the Minkowski, Minkowski-Blaschke, fiber, inner rotational, and outer rotational symmetrizations, are investigated. The focus is on the convergence of successive symmetrals with ... More
On convex bodies that are characterizable by volume functionAug 08 2019The "old-new" concept of convex-hull function was investigated by several authors in the last seventy years. A recent research on it led to some other volume functions as the covariogram function, the widthness function or the so-called brightness functions, ... More
Wasserstein stability of porous medium-type equations on manifolds with Ricci curvature bounded belowAug 08 2019Given a complete, connected Riemannian manifold $ \mathbb{M}^n $ with Ricci curvature bounded from below, we discuss the stability of the solutions of a porous medium-type equation with respect to the 2-Wasserstein distance. We produce (sharp) stability ... More
Two Phase Free Boundary Problem for Poisson KernelsAug 08 2019We provide a potential theoretic characterization of vanishing chord-arc domains under minimal assumptions. In particular we show that, in the appropriate class of domains, the oscillation of the logarithm of the interior and exterior Poisson kernels ... More
A hardness of approximation result in metric geometryAug 07 2019We show that it is $\mathsf{NP}$-hard to approximate the hyperspherical radius of a triangulated manifold up to an almost-polynomial factor.
Biangular Gabor frames and Zauner's conjectureAug 07 2019Two decades ago, Zauner conjectured that for every dimension $d$, there exists an equiangular tight frame consisting of $d^2$ vectors in $\mathbb{C}^d$. Most progress to date explicitly constructs the promised frame in various dimensions, and it now appears ... More
Some comments on Laakso graphs and sets of differencesAug 07 2019We recall a variation of a construction due to Laakso \cite{LA}, also used by Lang and Plaut \cite{LA} of a doubling metric space $X$ that cannot be embedded into any Hilbert space. We give a more concrete version of this construction and motivated by ... More
The largest angle bisection procedureAug 06 2019For a given triangle $\Delta ABC$, with $\angle A\ge \angle B \ge \angle C$, the $largest\, angle \, bisection\, procedure$ consists in constructing $AD$, the angle bisector of angle $\angle A$, and replacing $\Delta ABC$ by the two newly formed triangles, ... More
Grid dissections of tangential quadrilateralsAug 06 2019For any integer $n\ge 2$, a square can be partitioned into $n^2$ smaller squares via a checkerboard-type dissection. Does there such a class-preserving grid dissection exist for some other types of quadrilaterals? For instance, is it true that a tangential ... More
Outer linear measure of connected sets via Steiner treesAug 06 2019We resurrect an old definition of the linear measure of a metric continuum in terms of Steiner trees, independently due to Menger (1930) and Choquet (1938). We generalise it to any metric space and provide a proof of a little-known theorem of Choquet ... More
Rough $I$-convergence in cone metric spacesAug 06 2019Here we have studied the notion of rough $I$-convergence as an extension of the idea of rough convergence in a cone metric space using ideals. We have further introduced the notion of rough $I^*$-convergence of sequences in a cone metric space to find ... More
Lutwak-Petty projection inequalities for Minkowski valuations and their dualsAug 05 2019Lutwak's volume inequalities for polar projection bodies of all orders are generalized to polarizations of Minkowski valuations generated by even, zonal measures on the Euclidean unit sphere. This is based on analogues of mixed projection bodies for such ... More
The Quasi Curvature-Dimension Condition with applications to sub-Riemannian manifoldsAug 05 2019Aug 18 2019We introduce a quasi-convex relaxation of the $\mathsf{CD}(K,N)$ condition we call the Quasi Curvature-Dimension condition $\mathsf{QCD}(Q,K,N)$. Our motivation stems from a recent interpolation inequality along Wasserstein geodesics in the ideal sub-Riemannian ... More
The Quasi Curvature-Dimension Condition with applications to sub-Riemannian manifoldsAug 05 2019We introduce a quasi-convex relaxation of the $\mathsf{CD}(K,N)$ condition we call the Quasi Curvature-Dimension condition $\mathsf{QCD}(Q,K,N)$. Our motivation stems from a recent interpolation inequality along Wasserstein geodesics in the ideal sub-Riemannian ... More
The Quasi Curvature-Dimension Condition with applications to sub-Riemannian manifoldsAug 05 2019Aug 07 2019We introduce a quasi-convex relaxation of the $\mathsf{CD}(K,N)$ condition we call the Quasi Curvature-Dimension condition $\mathsf{QCD}(Q,K,N)$. Our motivation stems from a recent interpolation inequality along Wasserstein geodesics in the ideal sub-Riemannian ... More
Relationships Between Six IncirclesAug 04 2019If P is a point inside triangle ABC, then the cevians through P divide triangle ABC into six smaller triangles. We give theorems about the relationship between the radii of the circles inscribed in these triangles.
On Extensions of the Loomis-Whitney Inequality and Ball's Inequality for Concave, Homogeneous MeasuresAug 04 2019The Loomis-Whitney inequality states that the volume of a convex body is bounded by the product of volumes of its projections onto orthogonal hyperplanes. We provide a extension of both this fact and a generalization of this fact due to Ball to the context ... More
An extension of Berwald's inequality and its relation to Zhang's inequalityAug 03 2019In this note prove the following Berwald-type inequality, showing that for any integrable log-concave function $f:\mathbb R^n\rightarrow[0,\infty)$ and any concave function $h:L\rightarrow\mathbb [0,\infty)$, where $L$ is the epigraph of $-\log \frac{f}{\Vert ... More
Rational Elliptic Surfaces and the Trigonometry of TetrahedraAug 03 2019We study the trigonometry of non-Euclidean tetrahedra using tools from algebraic geometry. We establish a bijection between non-Euclidean tetrahedra and certain rational elliptic surfaces. We interpret the edge lengths and the dihedral angles of a tetrahedron ... More
Multiexponential maps in Carnot groups with applications to convexity and differentiabilityAug 03 2019We discuss a class of multiexponential maps in Carnot groups. We introduce a notion of multiexponential regularity and we show that such condition ensures a "cone property" for horizontally convex sets. Furthermore, we show that multiexponential regularity ... More
Optimal measures for p-frame energies on spheresAug 02 2019We provide new answers about the placement of mass on spheres so as to minimize energies of pairwise interactions. We find optimal measures for the $p$-frame energies, i.e. energies with the kernel given by the absolute value of the inner product raised ... More
On Theorems of Sinajova, Rankin and Kuperberg Concerning Spherical Point ConfigurationsAug 02 2019This note presents simple linear algebraic proofs of theorems due to Sinajova, Rankin and Kuperberg concerning spherical point configurations. The common ingredient in these proofs is the use of spherical Euclidean distance matrices and the Perron-Frobenius ... More
On Theorems of Sinajova, Rankin and Kuperberg Concerning Spherical Point ConfigurationsAug 02 2019Aug 16 2019This note presents simple linear algebraic proofs of theorems due to Sinajova, Rankin and Kuperberg concerning spherical point configurations. The common ingredient in these proofs is the use of spherical Euclidean distance matrices and the Perron-Frobenius ... More
Fast convergence of empirical barycenters in Alexandrov spaces and the Wasserstein spaceAug 02 2019This work establishes fast rates of convergence for empirical barycenters over a large class of geodesic spaces with curvature bounds in the sense of Alexandrov. More specifically, we show that parametric rates of convergence are achievable under natural ... More
On the perimeter length determination of the eight-centered ovalAug 02 2019On the perimeter length determination of the eight-centered oval. Several studies have shown that an eight-centered oval coincides almost perfectly with the ellipse constructed on the same axes and can be considered as a representation of the latter provided ... More
$L^1$-Monge problem in metric spaces possibly with branching geodesicsAug 02 2019In this paper, we consider the Monge optimal transport problem with distance cost. We prove that in some metric spaces, possibly with many branching geodesics, an optimal transport map exists if the first marginal is absolutely continuous. The result ... More
Pure point measures with sparse support and sparse Fourier--Bohr supportAug 01 2019Fourier transformable Radon measures are called doubly sparse when both the measure and its transform are pure point measures withsparse support. Their structure is reasonably well understood in Euclidean space, based on the use of tempered distributions. ... More
On the existence of paradoxical motions of generically rigid graphs on the sphereAug 01 2019We interpret realizations of a graph on the sphere up to rotations as elements of a moduli space of curves of genus zero. We focus on those graphs that admit an assignment of edge lengths on the sphere resulting in a flexible object. Our interpretation ... More
A metric space with its transfinite asymptotic dimension omega + 1Aug 01 2019We construct a metric space whose transfinite asymptotic dimension and complementary-finite asymptotic dimension are both omega+1, where omega is the smallest infinite ordinal number. Therefore, we prove that the omega conjecture is not true.
Continued fractions in non-Euclidean imaginary quadratic fieldsJul 31 2019Aug 02 2019In the Euclidean imaginary quadratic fields, continued fractions have been used to give rational approximations to complex numbers since the late 19th century. A variety of algorithms have been proposed in the 130 years following their introduction, but ... More
Continued fractions in non-Euclidean imaginary quadratic fieldsJul 31 2019In the Euclidean imaginary quadratic fields, continued fractions have been used to give rational approximations to complex numbers since the late 19\textsuperscript{th} century. A variety of algorithms have been proposed in the 130 years following their ... More
Binary Component Decomposition Part I: The Positive-Semidefinite CaseJul 31 2019This paper studies the problem of decomposing a low-rank positive-semidefinite matrix into symmetric factors with binary entries, either $\{\pm 1\}$ or $\{0,1\}$. This research answers fundamental questions about the existence and uniqueness of these ... More
Binary component decomposition Part II: The asymmetric caseJul 31 2019This paper studies the problem of decomposing a low-rank matrix into a factor with binary entries, either from $\{\pm 1\}$ or from $\{0,1\}$, and an unconstrained factor. The research answers fundamental questions about the existence and uniqueness of ... More
An Elastic Energy Minimization Framework for Mean Surface CalculationJul 31 2019As the continuation of the contour mean calculation - designed for averaging the manual delineations of 3D layer stack images - in this paper, the most important equations: a) the reparameterization equations to determine the minimizing diffeomorphism ... More
Sharp Stability of Brunn-Minkowski for Homothetic RegionsJul 30 2019The Brunn-Minkowski inequality applied to homothetic regions states that $|A| \le |tA+(1-t)A|$ for $A\subset \mathbb{R}^n$ and $t \in [0,1]$. We show there is a constant $C_n>0$ and constants $d_n(\tau)>0$ for each $\tau \in (0,\frac{1}{2}]$ such that ... More
Smale endomorphisms over graph-directed Markov systemsJul 30 2019In this paper we study Smale skew product endomorphisms (introduced in [21]) now over countable graph directed Markov systems, and we prove the exact dimensionality of conditional measures in fibers, and then the global exact dimensionality of the equilibrium ... More
Equiangular lines with a fixed angleJul 29 2019Solving a longstanding problem on equiangular lines, we determine, for each given fixed angle and in all sufficiently large dimensions, the maximum number of lines pairwise separated by the given angle. Fix $0 < \alpha < 1$. Let $N_\alpha(d)$ denote the ... More
Equiangular lines with a fixed angleJul 29 2019Aug 04 2019Solving a longstanding problem on equiangular lines, we determine, for each given fixed angle and in all sufficiently large dimensions, the maximum number of lines pairwise separated by the given angle. Fix $0 < \alpha < 1$. Let $N_\alpha(d)$ denote the ... More
Inverse problem for Moebius geometry on the circleJul 29 2019We give a solution to the inverse problem of Moebius geometry on the circle. Namely, we describe a class of Moebius structures on the circle for each of which there is a hyperbolic space such that its boundary at infinity is the circle, and the induced ... More
On the extendability by continuity of angular valuations on polytopesJul 26 2019A classical theorem of P. McMullen describes all valuations on polytopes that are invariant under translations and weakly continuous, i.e., continuous with respect to parallel displacements of the facets of a polytope. While it is typically not difficult ... More
On the extendability by continuity of angular valuations on polytopesJul 26 2019Aug 14 2019A classical theorem of P. McMullen describes all valuations on polytopes that are invariant under translations and weakly continuous, i.e., continuous with respect to parallel displacements of the facets of a polytope. While it is typically not difficult ... More
Rational Motions with Generic Trajectories of Low DegreeJul 26 2019The trajectories of a rational motion given by a polynomial of degree n in the dual quaternion model of rigid body displacements are generically of degree 2n. In this article we study those exceptional motions whose trajectory degree is lower. An algebraic ... More
Geometry and topology of symmetric point arrangementsJul 25 2019We investigate point arrangements $v_i\in\mathbb R^d,i\in \{1,...,n \}$ with certain prescribed symmetries. The arrangement space of $v$ is the column span of the matrix in which the $v_i$ are the rows. We characterize properties of $v$ in terms of the ... More
Fourier transform of Rauzy fractals and point spectrum of 1D Pisot inflation tilingsJul 25 2019Primitive inflation tilings of the real line with finitely many tiles of natural length and a Pisot-Vijayaraghavan unit as inflation factor are considered. We present an approach to the pure point part of their diffraction spectrum on the basis of a Fourier ... More
Curvature of the space of stability conditionsJul 25 2019Motivated by study of the autoequivalence group of triangulated categories via isometric actions on metric spaces, we consider curvature properties (CAT(0), Gromov hyperbolic) of the space of Bridgeland stability conditions with the canonical metric defined ... More
Convex sequences for Minkowski sums shorten the proof of a reverse isoperimetric inequalityJul 25 2019Recently, Chernov et al. [Adv. Math. 353 (2019)] proved that a sausage body is the unique solution to a reverse isoperimetric inequality within the class of $\lambda$-concave convex bodies, which are a particular type of Minkowski sums. We extend those ... More
Convex sequences for Minkowski sums shorten the proof of a reverse isoperimetric inequalityJul 25 2019Jul 30 2019Recently, Chernov et al. [Adv. Math. 353 (2019)] proved that a sausage body is the unique solution to a reverse isoperimetric inequality within the class of $\lambda$-concave convex bodies, which are a particular type of Minkowski sums. We extend those ... More
Indecomposable sets of finite perimeter in doubling metric measure spacesJul 25 2019We study a measure-theoretic notion of connectedness for sets of finite perimeter in the setting of doubling metric measure spaces supporting a weak $(1,1)$-Poincar\'{e} inequality. The two main results we obtain are a decomposition theorem into indecomposable ... More
The minimizers of the $p$-frame potentialJul 25 2019For any positive real number $p$, the $p$-frame potential of $N$ unit vectors $X:=\{\mathbf x_1,\ldots,\mathbf x_N\}\subset \mathbb R^d$ is defined as ${\rm FP}_{p,N,d}(X)=\sum_{i\neq j}|\langle \mathbf x_i,\mathbf x_j\rangle |^p$. In this paper, we focus ... More
The minimizers of the $p$-frame potentialJul 25 2019Aug 15 2019For any positive real number $p$, the $p$-frame potential of $N$ unit vectors $X:=\{\mathbf x_1,\ldots,\mathbf x_N\}\subset \mathbb R^d$ is defined as ${\rm FP}_{p,N,d}(X)=\sum_{i\neq j}|\langle \mathbf x_i,\mathbf x_j\rangle |^p$. In this paper, we focus ... More
The minimizers of the $p$-frame potentialJul 25 2019Jul 30 2019For any positive real number $p$, the $p$-frame potential of $N$ unit vectors $X:=\{\mathbf x_1,\ldots,\mathbf x_N\}\subset \mathbb R^d$ is defined as ${\rm FP}_{p,N,d}(X)=\sum_{i\neq j}|\langle \mathbf x_i,\mathbf x_j\rangle |^p$. In this paper, we focus ... More
The minimizers of the $p$-frame potentialJul 25 2019Aug 01 2019For any positive real number $p$, the $p$-frame potential of $N$ unit vectors $X:=\{\mathbf x_1,\ldots,\mathbf x_N\}\subset \mathbb R^d$ is defined as ${\rm FP}_{p,N,d}(X)=\sum_{i\neq j}|\langle \mathbf x_i,\mathbf x_j\rangle |^p$. In this paper, we focus ... More
Coarse Baum-Connes conjecture and rigidity for Roe algebrasJul 24 2019In this paper, we connect the rigidity problem and the coarse Baum-Connes conjecture for Roe algebras. In particular, we show that if $X$ and $Y$ are two uniformly locally finite metric spaces such that their Roe algebras are $*$-isomorphic, then $X$ ... More
How support lines touch an arcJul 24 2019We prove that each simple polygonal arc {\gamma} attains at most two pairs of support lines of given angle difference such that each pair has s1 < s2 < s3 that {\gamma}(s1) and {\gamma}(s3) are on one such line and {\gamma}(s2) is on the other line.
An aperiodic tile with edge-to-edge orientational matching rulesJul 23 2019We present a single, connected tile which can tile the plane but only non-periodically. The tile is hexagonal with edge markings, which impose simple rules as to how adjacent tiles are allowed to meet across edges. The first of these rules is a standard ... More
Non-commutative groups as prescribed polytopal symmetriesJul 23 2019We study properties of the realizations of groups as the combinatorial automorphism group of a convex polytope. We show that for any non-abelian group $G$ with a central involution there is a centrally symmetric polytope with $G$ as its combinatorial ... More
Asymptotic Filtered ColimitsJul 23 2019If one has a collection of large scale spaces $\{(X_s,\mathcal{LSS}_s)\}_{s\in S}$ with certain compatibility conditions one may define a large scale space on $X=\bigcup\limits_{s\in S}X_s$ in a way where every function on $X$ is large scale continuous ... More
Hofer's metric in compact Lie groupsJul 23 2019In this article we study the Hofer geometry of a compact Lie group $K$ which acts by Hamiltonian diffeomorphisms on a symplectic manifold $M$. Generalized Hofer norms on the Lie algebra of $K$ are introduced and analyzed with tools from group invariant ... More
Molecules as metric measure spaces with Kato-bounded Ricci curvatureJul 22 2019Let $\Psi>0$ be the logarithm of the ground state of an arbitrary molecule with $n$ electrons in the infinite mass limit (neglecting spin/statistics). Let $\Sigma\subset \mathbb{R}^{3n}$ be the set of singularities of the underlying Coulomb potential. ... More
Molecules as metric measure spaces with Kato-bounded Ricci curvatureJul 22 2019Aug 01 2019Let $\Psi>0$ be the logarithm of the ground state of an arbitrary molecule with $n$ electrons in the infinite mass limit (neglecting spin/statistics). Let $\Sigma\subset \mathbb{R}^{3n}$ be the set of singularities of the underlying Coulomb potential. ... More
The Gromov--Hausdorff Distance between Simplexes and Two-Distance SpacesJul 22 2019In 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 ... More
An intrinsic characterization of five points in a $\mathrm{CAT}(0)$ spaceJul 22 2019We prove that a metric space that contains at most five points admits an isometric embedding into a $\mathrm{CAT}(0)$ space if and only if it satisfies the $\boxtimes$-inequalities.
An intrinsic characterization of five points in a $\mathrm{CAT}(0)$ spaceJul 22 2019Aug 18 2019Gromov (2001) and Sturm (2003) proved that any four points in a $\mathrm{CAT}(0)$ space satisfy a certain family of inequalities. We call those inequalities the $\boxtimes$-inequalities, following the notation used by Gromov. In this paper, we prove that ... More
A remark on Gromov's cycle conditionsJul 22 2019We prove that if a metric space $X$ satisfies Gromov's $\mathrm{Cycl}_4 (0)$ condition, or equivalently, if $X$ satisfies the $\boxtimes$-inequalities, then $X$ satisfies Gromov's $\mathrm{Cycl}_k (0)$ condition for every integer $k\geq 4$.
An improved constant in Banaszczyk's transference theoremJul 21 2019$ \newcommand{\R}{\ensuremath{\mathbb{R}}} \newcommand{\lat}{\mathcal{L}} \newcommand{\ensuremath}[1]{#1} $We show that \[ \mu(\lat) \lambda_1(\lat^*) < \big( 0.1275 + o(1) \big) \cdot n \; , \] where $\mu(\lat)$ is the covering radius of an $n$-dimensional ... More
Measuring the local non-convexity of real algebraic curvesJul 19 2019The goal of this paper is to measure the non-convexity of compact and smooth connected components of real algebraic plane curves. We study these curves first in a general setting and then in an asymptotic one. In particular, we consider sufficiently small ... More
Eigenfunctions of the Fourier Transform with specified zerosJul 19 2019We give a unified description of the modular and quasi-modular functions used in Viazovska's proof of the best packing bounds in dimension 8 and the proof by Cohn, Kumar, Miller, Radchenko, and Viazovska of the best packing bound in dimension 24. We show ... More
A note on the local Lipschitz triviality of values of complex polynomial functionsJul 19 2019We address the question of the bi-Lipschitz local triviality of a complex polynomial function over a complex value. Our main result state that a non constant complex polynomial admits a locally bi-Lipschitz trivial value if and only if it is a polynomial ... More
Favourite distances in 3-spaceJul 19 2019Let $S$ be a set of $n$ points in Euclidean $3$-space. Assign to each $x\in S$ a distance $r(x)>0$, and let $e_r(x,S)$ denote the number of points in $S$ at distance $r(x)$ from $x$. Avis, Erd\H{o}s and Pach (1988) introduced the extremal quantity $f_3(n)=\max\sum_{x\in ... More
A note on sets avoiding rational distancesJul 18 2019In this paper we shall give a short proof of the result originally obtained by Ashutosh Kumar that for each $A\subset \mathbb{R}$ there exists $B\subset A$ full in $A$ such that no distance between two distinct points from $B$ is rational. We will construct ... More
Game of Sloanes: Best known packings in complex projective spaceJul 18 2019It is often of interest to identify a given number of points in projective space such that the minimum distance between any two points is as large as possible. Such configurations yield representations of data that are optimally robust to noise and erasures. ... More
Projection theorems for intermediate dimensionsJul 17 2019Aug 15 2019Intermediate dimensions were recently introduced to interpolate between the Hausdorff and box-counting dimensions of fractals. Firstly, we show that these intermediate dimensions may be defined in terms of capacities with respect to certain kernels. Then, ... More
Projection theorems for intermediate dimensionsJul 17 2019Intermediate dimensions were recently introduced to interpolate between the Hausdorff and box-counting dimensions of fractals. Firstly, we show that these intermediate dimensions may be defined in terms of capacities with respect to certain kernels. Then, ... More
The box dimensions of exceptional self-affine sets in $\mathbb{R}^3$Jul 17 2019We study the box dimensions of self-affine sets in $\mathbb{R}^3$ which are generated by a finite collection of generalised permutation matrices. We obtain bounds for the dimensions which hold with very minimal assumptions and give rise to sharp results ... More