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On $BV$ functions and essentially bounded divergence-measure fields in metric spacesJun 18 2019By employing the differential structure recently developed by N. Gigli, we can extend the notion of divergence-measure vector fields $\mathcal{DM}^p(\mathbb{X})$, $1\le p \le \infty$, to the very general context of a (locally compact) metric measure space ... More

Combinatorial characterization of pseudometricsJun 18 2019Let $X$, $Y$ be sets and let $\Phi$, $\Psi$ be mappings with the domains $X^{2}$ and $Y^{2}$ respectively. We say that $\Phi$ is combinatorially similar to $\Psi$ if there are bijections $f \colon \Phi(X^2) \to \Psi(Y^{2})$ and $g \colon Y \to X$ such ... More

Random geometric complexes and graphs on Riemannian manifolds in the thermodynamic limitJun 17 2019We investigate some topological properties of random geometric complexes and random geometric graphs on Riemannian manifolds in the thermodynamic limit. In particular, for random geometric complexes we prove that the normalized counting measure of connected ... More

Associahedra for finite type cluster algebras and minimal relations between $\mathbf{g}$-vectorsJun 17 2019We show that the mesh mutations are the minimal relations among the $\mathbf{g}$-vectors with respect to any initial seed in any finite type cluster algebra. We then use this algebraic result to derive geometric properties of the $\mathbf{g}$-vector fan: ... More

On the Configuration Space of Steiner Minimal TreesJun 15 2019Among other results, we prove the following theorem about Steiner minimal trees in $d$-dimensional Euclidean space: if two finite sets in $\mathbb{R}^d$ have unique and combinatorially equivalent Steiner minimal trees, then there is a homotopy between ... More

The unique measure of maximal entropy for a compact rank one locally CAT(0) spaceJun 14 2019Let $X$ be a compact, geodesically complete, locally CAT(0) space such that the universal cover admits a rank one axis. We prove the Bowen-Margulis measure on the space of geodesics is the unique measure of maximal entropy for the geodesic flow, which ... More

Large Sets with Small Injective ProjectionsJun 14 2019Let $\ell_1,\ell_2,\dots$ be a countable collection of lines in ${\mathbb R}^d$. For any $t \in [0,1]$ we construct a compact set $\Gamma\subset{\mathbb R}^d$ with Hausdorff dimension $d-1+t$ which projects injectively into each $\ell_i$, such that the ... More

On inscribed equilateral simplices in normed spacesJun 14 2019In this paper we prove in certain n-dimensional normed spaces X the existence of full-dimensional equilateral simplices of large size inscribed to the unit ball B. This extends the construction of Makeev [Mak] in dimension 4 and we also compute an example ... More

Monotone vector fields and generation of nonexpansive semigroups in complete CAT(0) spacesJun 14 2019In this paper, we discuss about monotone vector fields, which is a typical extension to the theory of convex functions, by exploiting the tangent space structure. This new approach to monotonicity in CAT(0) spaces stands in opposed to the monotonicity ... More

Geometric averaging operators and nonconcentration inequalitiesJun 11 2019This paper is devoted to a systematic study of certain geometric integral inequalities which arise in continuum combinatorial approaches to $L^p$-improving inequalities for Radon-like transforms over polynomial submanifolds of intermediate dimension. ... More

Pure entropic regularization for metrical task systemsJun 10 2019We show that on every $n$-point HST metric, there is a randomized online algorithm for metrical task systems (MTS) that is $1$-competitive for service costs and $O(\log n)$-competitive for movement costs. In general, these refined guarantees are optimal ... More

Actions of solvable Baumslag-Solitar groups on hyperbolic metric spacesJun 10 2019We give a complete list of the cobounded actions of solvable Baumslag-Solitar groups on hyperbolic metric spaces up to a natural equivalence relation. The set of equivalence classes carries a natural partial order first introduced by Abbott-Balasubramanya-Osin, ... More

An Optimal Plank TheoremJun 10 2019It is shown that for any sequence $v_1,v_2,\dots,v_n$ of unit vectors in a real Hilbert space $H$, there exists a unit vector $v$ in $H$ such that $$|\langle v_k,v \rangle| \geq \sin(\pi/2n)$$ for all $k$. This a sharp version of the plank theorem for ... More

Dot product invariant valuations on Lip$(S^{n-1})$Jun 10 2019We provide an integral representation for continuous, rotation invariant and dot product invariant valuations defined on the space Lip$(S^{n-1})$ of Lipschitz continuous functions on the unit $n-$sphere.

Rips complexes as nerves and a Functorial Dowker-Nerve DiagramJun 10 2019Using ideas of the Dowker duality we prove that the Rips complex at scale $r$ is homotopy equivalent to the nerve of a cover consisting of sets of prescribed diameter. We then develop a functorial version of the Nerve theorem coupled with the Dowker duality, ... More

Norms of weighted sums of log-concave random vectorsJun 09 2019Let $C$ and $K$ be centrally symmetric convex bodies of volume $1$ in ${\mathbb R}^n$. We provide upper bounds for the multi-integral expression \begin{equation*}\|{\bf t}\|_{C^s,K}=\int_{C}\cdots\int_{C}\Big\|\sum_{j=1}^st_jx_j\Big\|_K\,dx_1\cdots dx_s\end{equation*} ... More

A note on norms of signed sums of vectorsJun 09 2019Our starting point is an improved version of a result of D. Hajela related to a question of Koml\'{o}s: we show that if $f(n)$ is a function such that $\lim\limits_{n\to\infty }f(n)=\infty $ and $f(n)=o(n)$, there exists $n_0=n_0(f)$ such that for every ... More

High-dimensional limit theorems for random vectors in $\ell_p^n$-balls. IIJun 09 2019In this article we prove three fundamental types of limit theorems for the $q$-norm of random vectors chosen at random in an $\ell_p^n$-ball in high dimensions. We obtain a central limit theorem, a moderate deviations as well as a large deviations principle ... More

Smocked Metric Spaces and their Tangent ConesJun 08 2019We introduce the notion of a smocked metric spaces and explore the balls and geodesics in a collection of different smocked spaces. We find their rescaled Gromov-Hausdorff limits and prove these tangent cones at infinity exist, are unique, and are normed ... More

Hypercontractivity, and Lower Deviation Estimates in Normed SpacesJun 07 2019We consider the problem of estimating probabilities of lower deviation $\mathbb P\{\|G\| \leqslant \delta \mathbb E\|G\|\}$ in normed spaces with respect to the Gaussian measure. These estimates occupy central role in the probabilistic study of high-dimensional ... More

A new Federer-type characterization of sets of finite perimeter in metric spacesJun 07 2019Federer's characterization states that a set $E\subset \mathbb{R}^n$ is of finite perimeter if and only if $\mathcal H^{n-1}(\partial^*E)<\infty$. Here the measure-theoretic boundary $\partial^*E$ consists of those points where both $E$ and its complement ... More

Next levels universal bounds for spherical codes: the Levenshtein framework liftedJun 07 2019We introduce a framework based on the Delsarte-Yudin linear programming approach for improving some universal lower bounds for the minimum energy of spherical codes of prescribed dimension and cardinality, and universal upper bounds on the maximal cardinality ... More

Interior angle sums of geodesic triangles in $S^2 \times R$ and $H^2 \times R$ geometriesJun 07 2019In the present paper we study $S^2 \times R$ and $H^2 \times R$ geometries, which are homogeneous Thurston 3-geometries. We analyse the interior angle sums of geodesic triangles in both geometries and prove, that in $S^2 \times R$ space it can be larger ... More

Nearly $k$-distance setsJun 06 2019We say that a set of points $S\subset \mathbb{R}^d$ is an $\varepsilon$-nearly $k$-distance set if there exist $1\le t_1\le \ldots\le t_k,$ such that the distance between any two distinct points in $S$ falls into $[t_1,t_1+\varepsilon]\cup\ldots\cup[t_k,t_k+\varepsilon]$. ... More

Assouad spectrum thresholds for some random constructionsJun 06 2019The Assouad dimension of a metric space determines its extremal scaling properties. The derived notion of the Assouad spectrum fixes relative scales by a scaling function to obtain interpolation behaviour between the quasi-Assouad and box-counting dimensions. ... More

Bi-Lipschitz embeddings of $SRA$-free spaces into Euclidean spacesJun 06 2019$SRA$-free spaces is a wide class of metric spaces including finite dimensional Alexandrov spaces of non-negative curvature, complete Berwald spaces of nonnegative flag curvature, Cayley Graphs of virtually abelian groups and doubling metric spaces of ... More

Asymptotic normality for random simplices and convex bodies in high dimensionsJun 06 2019Central limit theorems for the log-volume of a class of random convex bodies in $\mathbb{R}^n$ are obtained in the high-dimensional regime, that is, as $n\to\infty$. In particular, the case of random simplices pinned at the origin and simplices where ... More

Behavior of convex surfaces near ridge pointsJun 05 2019The aim of this paper is twofold. First, we cut off a part of a convex surface by a plane near a ridge point and characterize the limiting behavior of the surface measure in $S^2$ induced by this part of surface when the plane approaches the point. Second, ... More

On the embeddability of the family of countably branching trees into quasi-reflexive Banach spacesJun 05 2019In this note we extend to the quasi-reflexive setting the result of F. Baudier, N. Kalton and G. Lancien concerning the non-embeddability of the family of countably branching trees into reflexive Banach spaces whose Szlenk index and Szlenk index from ... More

A combinatorial criterion for macroscopic circles in planar triangulationsJun 04 2019Given a finite simple triangulation, we estimate the sizes of circles in its circle packing in terms of Cannon's vertex extremal length. Our estimates provide control over the size of the largest circle in the packing. We use them, combined with results ... More

The density and minimal gap of visible points in some planar quasicrystalsJun 04 2019We give formulas for the density of visible points of several families of planar quasicrystals, which include the Ammann-Beenker point set and vertex sets of some rhombic Penrose tilings. These densities are used in order to calculate the limiting minimal ... More

Induction of $\mathbb{Z}^2$-actions and of partitions of the 2-torusJun 03 2019Sturmian sequences are the most simple aperiodic sequences. A result of Morse, Hedlund (1940) and Coven, Hedlund (1970) is that a biinfinite binary sequence is sturmian if and only if it is obtained as the coding of an irrational rotation on the circle ... More

Generalizations of the Drift Laplace Equation in the Heisenberg Group and a Class of Grushin-Type SpacesJun 03 2019We find fundamental solutions to p-Laplace equations with drift terms in the Heisenberg group and Grushin-type planes. These solutions are natural generalizations to the fundamental solutions discovered by Beals, Gaveau, and Greiner for the Laplace equation ... More

Pairing symmetries for Euclidean and spherical frameworksJun 03 2019Jun 06 2019In this paper we consider the effect of symmetry on the rigidity of bar-joint frameworks, spherical frameworks and point-hyperplane frameworks in $\mathbb{R}^d$. In particular we show that, under forced or incidental symmetry, infinitesimal rigidity for ... More

Pairing symmetries for projective and spherical frameworksJun 03 2019In this paper we consider the effect of symmetry on the rigidity of bar-joint frameworks, spherical frameworks and point-hyperplane frameworks in $\mathbb{R}^d$. In particular we show that, under forced or incidental symmetry, infinitesimal rigidity for ... More

Sets in $\mathbb{R}^d$ with slow-decaying density that avoid an unbounded collection of distancesJun 02 2019Jun 05 2019For any $d\in \mathbb{N}$ and any function $f:(0,\infty)\to [0,1]$ with $f(R)\to 0$ as $R\to \infty$, we construct a set $A \subseteq \mathbb{R}^d$ and a sequence $R_n \to \infty$ such that $\|x-y\| \neq R_n$ for all $x,y\in A$ and $\mu(A\cap B_{R_n})\geq ... More

The Scott rank of Polish metric spacesJun 02 2019We study the usual notion of Scott rank but in the setting of Polish metric spaces. The signature consists of distance relations: for each rational $q > 0$, there is a relation $R_{<q}(x,y)$ stating that the distance of $x$ and $y $ is less than $q$. ... More

Vertical versus horizontal Sobolev spacesMay 31 2019Let $\alpha \geq 0$, $1 < p < \infty$, and let $\mathbb{H}^{n}$ be the Heisenberg group. Folland in 1975 showed that if $f \colon \mathbb{H}^{n} \to \mathbb{R}$ is a function in the horizontal Sobolev space $S^{p}_{2\alpha}(\mathbb{H}^{n})$, then $\varphi ... More

Heat kernels are not uniform expandersMay 31 2019We study infinite analogues of expander graphs, namely graphs where subgraphs weighted by heat kernels form an expander family. Our main result is that there does not exist any infinite expander in this sense. This proves the analogue for random walks ... More

A brute force computer aided proof of an existence result about extremal hyperbolic surfacesMay 30 2019Extremal compact hyperbolic surfaces contain a packing of discs of the largest possible radius permitted by the topology of the surface. It is well known that arithmetic conditions on the uniformizing group are necessary for the existence of a second ... More

Extremal k-packings in compact non-orientable surfacesMay 30 2019An extremal $k$-packing is a collection of $k$ mutually disjoint metric discs, embedded in a surface, whose radius is maximal for the given topology. We study compact non-orientable surfaces of genus $g\ge 3$ containing extremal $k$-packings.

Largest Inscribed Rectangles in Geometric Convex SetsMay 30 2019We consider the problem of finding inscribed boxes and axis-aligned inscribed boxes of maximum volume, inside a compact and solid convex set. Our algorithms are capable of solving these two problems in any such set that can be represented with finite ... More

From Hierarchical to Relative HyperbolicityMay 29 2019We provide a simple, combinatorial criteria for a hierarchically hyperbolic space to be relatively hyperbolic by proving a new formulation of relative hyperbolicity in terms of hierarchy structures. In the case of clean hierarchically hyperbolic groups, ... More

A rational trigonometric relationship between the dihedral angles of a tetrahedron and its circumradiusMay 29 2019Jun 12 2019This paper will extend a known relationship between the circumradius and dihedral angles of a tetrahedron in three-dimensional Euclidean space to three-dimensional affine space over a general field not of characteristic two, using only the framework of ... More

A rational trigonometric relationship between the dihedral angles of a tetrahedron and its circumradiusMay 29 2019This paper will derive an analog of a known relationship between the circumradius and dihedral angles of a tetrahedron in three-dimensional Euclidean space, using only the framework of rational trigonometry that was devised by Wildberger in 2005. Such ... More

The sharp $p$-Poincaré inequality under the measure contraction propertyMay 28 2019We obtain sharp estimate on $p$-spectral gaps, or equivalently optimal constant in $p$-Poincar\'e inequalities, for metric measure spaces satisfying measure contraction property. We also prove the rigidity for the sharp $p$-spectral gap.

Chasing Convex Bodies OptimallyMay 28 2019In the chasing convex bodies problem, an online player receives a request sequence of $N$ convex sets $K_1,\dots, K_N$ contained in a normed space $\mathbb R^d$. The player starts at $x_0\in \mathbb R^d$, and after observing each $K_n$ picks a new point ... More

Chasing Convex Bodies with Linear Competitive RatioMay 28 2019We study the problem of chasing convex bodies online: given a sequence of convex bodies $K_t\subseteq \mathbb{R}^d$ the algorithm must respond with points $x_t\in K_t$ in an online fashion (i.e., $x_t$ is chosen before $K_{t+1}$ is revealed). The objective ... More

Inverse Blaschke-Santaló inequality for convex curves enclosing the origin several timesMay 28 2019H. Guggenheimer generalized the planar volume product problem for locally convex curves $C$ enclosing the origin $k \ge 2$ times. He conjectured that the minimal volume product $V(C)V(C^*)$ for these curves is attained if the curve consists of the longest ... More

Interpolating between dimensionsMay 27 2019Dimension theory lies at the heart of fractal geometry and concerns the rigorous quantification of how large a subset of a metric space is. There are many notions of dimension to consider, and part of the richness of the subject is in understanding how ... More

Tensegrities on the space of generic functionsMay 27 2019In this small note we introduce a notion of self-stresses on the set functions in two variables with generic critical points. The notion naturally comes from a rather exotic representation of classical Maxwell frameworks in terms of differential forms. ... More

On the lattice Hadwiger number of superballs and some other bodiesMay 27 2019May 29 2019We show that the lattice Hadwiger number of superballs is exponential in the dimension. The same is true for some more general convex bodies.

On the lattice Hadwiger number of superballs and some other bodiesMay 27 2019Jun 05 2019We show that the lattice Hadwiger number of superballs is exponential in the dimension. The same is true for some more general convex bodies.

Extra large type Artin groups are CAT(0) and acylindrically hyperbolicMay 27 2019We describe a simple locally CAT(0) classifying space for extra large type Artin groups. Furthermore, when the Artin group is not dihedral, we describe a rank 1 periodic geodesic, thus proving that extra large type Artin groups are acylindrically hyperbolic. ... More

Extra large type Artin groups are CAT(0) and acylindrically hyperbolicMay 27 2019Jun 07 2019We describe a simple locally CAT(0) classifying space for extra large type Artin groups. Furthermore, when the Artin group is not dihedral, we describe a rank 1 periodic geodesic, thus proving that extra large type Artin groups are acylindrically hyperbolic. ... More

Geodesics in persistence diagram spaceMay 26 2019It is known that for a variety of choices of metrics, including the standard bottleneck distance, the space of persistence diagrams admits geodesics. Typically these existence results produce geodesics that have the form of a convex combination. More ... More

Covering by homothets and illuminating convex bodiesMay 25 2019The paper is devoted to coverings by translative homothets and illuminations of convex bodies. For a given positive number $\alpha$ and a convex body $B$, $g_{\alpha}(B)$ is the infimum of $\alpha$-powers of finitely many homothety coefficients less than ... More

Covering by homothets and illuminating convex bodiesMay 25 2019Jun 02 2019The paper is devoted to coverings by translative homothets and illuminations of convex bodies. For a given positive number $\alpha$ and a convex body $B$, $g_{\alpha}(B)$ is the infimum of $\alpha$-powers of finitely many homothety coefficients less than ... More

Classification of uniformly distributed measures of dimension $1$ in general codimensionMay 23 2019Starting with the work of Preiss on the geometry of measures, the classification of uniform measures in $\mathbb R^d$ has remained open, except for $d=1$ and for compactly supported measures in $d=2$, and for codimension $1$. In this paper we study $1$-dimensional ... More

Escaping a neighborhood along a prescribed sequence in Lie groups and Banach algebrasMay 23 2019It is shown that Jamison sequences, introduced in [C. Badea and S. Grivaux, Unimodular eigenvalues, uniformly distributed sequences and linear dynamics, Adv. Math. 211 (2007), no. 2, 766--793], arise naturally in the study of topological groups with no ... More

The Space of Persistence Diagrams on $n$ Points Coarsely Embeds into Hilbert SpaceMay 22 2019We prove that the space of persistence diagrams on $n$ points (with the bottleneck or a Wasserstein distance) coarsely embeds into Hilbert space by showing it is of asymptotic dimension $2n$. Such an embedding enables utilisation of Hilbert space techniques ... More

Constant diameter and constant width of spherical convex bodiesMay 22 2019In this note we show that a spherical convex body $C$ is of constant diameter $a$ if and only if $C$ is of constant width $a$, for $0<a<\pi$.

Harmonic Measure and the Analyst's Traveling Salesman TheoremMay 22 2019We study how generalized Jones $\beta$-numbers relate to harmonic measure. Firstly, we generalize a result of Garnett, Mourgoglou and Tolsa by showing that domains in $\mathbb{R}^{d+1}$ whose boundaries are lower $d$-content regular admit Corona decompositions ... More

Sublinear quasiconformality and the large-scale geometry of Heintze groupsMay 22 2019This article analyzes sublinearly quasisymmetric homeo-morphisms (generalized quasisymmetric mappings), and draws applications to the sublinear large-scale geometry of negatively curved groups and spaces. It is proven that those homeomorphisms lack analytical ... More

Intrinsic and dual volume deviations of convex bodies and polytopesMay 21 2019We establish estimates for the asymptotic best approximation of the Euclidean unit ball by polytopes under a notion of distance induced by the intrinsic volumes. We also introduce a notion of distance between convex bodies that is induced by the Wills ... More

Convexity in Greek antiquityMay 21 2019We consider several appearances of the notion of convexity in Greek antiquity, more specifically in mathematics and optics, in the writings of Aristotle, and in art. The final version of this article will appear in the book `Geometry in History', ed. ... More

Geodesic rays, the "Lion-Man" game, and the fixed point propertyMay 20 2019This paper focuses on the relation among the existence of different types of curves (such as directional ones, quasi-geodesic or geodesic rays), the (approximate) fixed point property for nonexpansive mappings, and a discrete lion and man game. Our main ... More

A strongly irreducible affine iterated function system with two invariant measures of maximal dimensionMay 20 2019A classical theorem of Hutchinson asserts that if an iterated function system acts on $\mathbb{R}^d$ by similitudes and satisfies the open set condition then it admits a unique self-similar measure with Hausdorff dimension equal to the dimension of the ... More

Incongruent equipartitions of the planeMay 20 2019R. Nandakumar asked whether there is a tiling of the plane by pairwise incongruent triangles of equal area and equal perimeter. Recently a negative answer was given by Kupavskii, Pach and Tardos. Still one may ask for weaker versions of the problem, or ... More

Sharp estimate on the inner distance in planar domainsMay 20 2019We show that the inner distance inside a bounded planar domain is at most the one-dimensional Hausdorff measure of the boundary of the domain. We prove this sharp result by establishing an improved Painlev\'e length estimate for connected sets and by ... More

Plus minus analogues for affine Tverberg type resultsMay 19 2019The classical 1966 theorem of Tverberg with its numerous variations was and still is a motivating force behind many important developments in convex and computational geometry as well as the testing ground for methods from equivariant algebraic topology. ... More

On Hölder solutions to the spiral winding problemMay 18 2019The winding problem concerns understanding the regularity of functions which map a line segment onto a spiral. This problem has relevance in fluid dynamics and conformal welding theory, where spirals arise naturally. Here we interpret `regularity' in ... More

Transport and Interface: an Uncertainty Principle for the Wasserstein distanceMay 17 2019May 21 2019Let $f: [0,1]^d \rightarrow \mathbb{R}$ be a continuous function with zero mean and interpret $f_{+} = \max(f, 0)$ and $f_{-} = -\min(f, 0)$ as the densities of two measures. We prove that if the cost of transport from $f_{+}$ to $f_{-}$ is small (in ... More

Transport and Interace: an Uncertainty Principle for the Wasserstein distanceMay 17 2019Let $f: [0,1]^d \rightarrow \mathbb{R}$ be a continuous function with zero mean and interpret $f_{+} = \max(f, 0)$ and $f_{-} = -\min(f, 0)$ as the densities of two measures. We prove that if the cost of transport from $f_{+}$ to $f_{-}$ is small (in ... More

Diameters of Ball IntersectionsMay 17 2019We prove the diameter of the intersection of two closed convex balls in a Riemannian manifold eventually decreases continuously as the centers of the balls move apart.

On generalized median triangles and tracing orbitsMay 17 2019Jun 06 2019We study generalization of median triangles on the plane with two complex parameters. By specialization of the parameters, we produce periodical motion of a triangle whose vertices trace each other on a common closed orbit.

On generalized median triangles and tracing orbitsMay 17 2019We study a generalization of median triangles on the plane with two complex parameters. By specialization of the parameters, we produce periodical motion of a triangle whose vertices trace each other on a common closed orbit.

Moebius automorphisms of surfaces with many circlesMay 16 2019We classify real two-dimensional orbits of conformal subgroups such that the orbits contain two circular arcs through a point. Such surfaces must be toric and admit a Moebius automorphism group of dimension at least two. Our theorem generalizes the classical ... More

Filling metric spacesMay 16 2019We prove an inequality conjectured by Larry Guth that relates the $m$-dimensional Hausdorff content of a compact metric space with its $(m-1)$-dimensional Urysohn width. As a corollary, we obtain new systolic inequalities that both strengthen the classical ... More

When a spherical body of constant diameter is of constant width?May 15 2019{\bf Abstract.} Let $D$ be a convex body of diameter $\delta$, where $0 < \delta < \frac{\pi}{2}$, on the $d$-dimensional sphere. We prove that $D$ is of constant diameter $\delta$ if and only if it is of constant width $\delta$ in the following two cases. ... More

Doubly transitive lines II: Almost simple symmetriesMay 15 2019We study lines through the origin of finite-dimensional complex vector spaces that enjoy a doubly transitive automorphism group. This paper, the second in a series, classifies those lines that exhibit almost simple symmetries. To perform this classification, ... More

On the isometric conjecture of BanachMay 14 2019May 28 2019We show that a real Banach space of dimension $N=4k+2\geq 6,$ $N\neq 134$, all of whose codimension 1 subspaces are isometrically isomorphic to each other, is a Hilbert space. This gives a partial answer to a conjecture of Stefan Banach from 1932.

On the isometric conjecture of BanachMay 14 2019We show that a real Banach space of dimension $N=4k+2\geq 6,$ $N\neq 134$, all of whose codimension 1 subspaces are isometrically isomorphic to each other, is a Hilbert space. This gives a partial answer to a conjecture of Stefan Banach from 1932.

1st eigenvalue pinching for convex hypersurfaces in a Riemannian manifoldMay 14 2019Let $M^n$ be a closed convex hypersurface lying in a convex ball $B(p,R)$ of the ambient $(n+1)$-manifold $N^{n+1}$. We prove that, by pinching Heintze-Reilly's inequality via sectional curvature upper bound of $B(p,R)$, 1st eigenvalue and mean curvature ... More

Busemann functions on the Wasserstein spaceMay 14 2019We study rays and co-rays in the Wasserstein space $P_p(\mathcal{X})$ ($p > 1$) whose ambient space $\mathcal{X}$ is a complete, separable, non-compact, locally compact length space. We show that rays in the Wasserstein space can be represented as probability ... More

A fibration theorem for collapsing sequences of Alexandrov spacesMay 14 2019Let a sequence $M_j$ of Alexandrov spaces collapse to a space $X$ with only weak singularities. T. Yamaguchi constructed a map $f_j:M_j\to X$ for large $j$ called an almost Lipschitz submersion. We prove that if $M_j$ has a uniform positive lower bound ... More

Regular points of extremal subsets in Alexandrov spacesMay 14 2019We define regular points of an extremal subset in an Alexandrov space and study their basic properties. We show that a neighborhood of a regular point in an extremal subset is almost isometric to an open subset in the Euclidean space and that the set ... More

Sharp Poincaré inequality under Measure Contraction PropertyMay 14 2019We prove a sharp Poincar\'e inequality for subsets $\Omega$ of (essentially non-branching) metric measure spaces satisfying the Measure Contraction Property $\textrm{MCP}(K,N)$, whose diameter is bounded above by $D$. This is achieved by identifying the ... More

Uniformizing surfaces via discrete harmonic mapsMay 14 2019We show that for any closed surface of genus greater than one and for any finite weighted graph filling the surface, there exists a hyperbolic metric which realizes the least Dirichlet energy harmonic embedding of the graph among a fixed homotopy class ... More

Homogenisation of one-dimensional discrete optimal transportMay 13 2019This paper deals with dynamical optimal transport metrics defined by discretisation of the Benamou--Benamou formula for the Kantorovich metric $W_2$. Such metrics appear naturally in discretisations of $W_2$-gradient flow formulations for dissipative ... More

Variational formulas for submanifolds of fixed degreeMay 13 2019We consider in this paper an area functional defined on submanifolds of fixed degree immersed into a graded manifold equipped with a Riemannian metric. Since the expression of this area depends on the degree, not all variations are admissible. It turns ... More

Approximate arithmetic structure in large sets of integersMay 13 2019We prove that if a set is `large' in the sense of Erd\H{o}s, then it approximates arbitrarily long arithmetic progressions in a strong quantitative sense. More specifically, expressing the error in the approximation in terms of the gap length $\Delta$ ... More

Visibility of Cartesian products of Cantor setsMay 12 2019Let $K_{\lambda}$ be the attractor of the following IFS \begin{equation*} \{f_1(x)=\lambda x, f_2(x)=\lambda x+1-\lambda\}, \;\;0<\lambda<1/2. \end{equation*} Given $\alpha \geq 0$, we say the line $y=\alpha x$ is visible through $K_{\lambda}\times K_{\lambda}$ ... More

Approximating a Target Surface with 1-DOF Rigid OrigamiMay 12 2019We develop some design examples for approximating a target surface at the final rigidly folded state of a developable quadrilateral creased paper, which is folded with a 1-DOF rigid folding motion from the planar state. The final rigidly folded state ... More

The diameter of lattice zonotopesMay 12 2019We establish sharp asymptotic estimates for the diameter of primitive zonotopes when their dimension is fixed. We also prove that, for infinitely many integers $k$, the largest possible diameter of a lattice zonotope contained in the hypercube $[0,k]^d$ ... More

Arithmetic on Moran setsMay 12 2019Let $(\mathcal{M}, c_k,n_k)$ be a class of Moran sets. We assume that the convex hull of any $E\in (\mathcal{M}, c_k,n_k)$ is $[0,1]$. Let $A,B$ be two non-empty sets in $\mathbb{R}$. Suppose that $f$ is a continuous function defined on an open set $U\subset ... More

Embeddings of Persistence Diagrams into Hilbert SpacesMay 11 2019Since persistence diagrams do not admit an inner product structure, a map into a Hilbert space is needed in order to use kernel methods. It is natural to ask if such maps necessarily distort the metric on persistence diagrams. We show that persistence ... More

Embeddings of Persistence Diagrams into Hilbert SpacesMay 11 2019May 27 2019Since persistence diagrams do not admit an inner product structure, a map into a Hilbert space is needed in order to use kernel methods. It is natural to ask if such maps necessarily distort the metric on persistence diagrams. We show that persistence ... More

Scaling of diffraction intensities near the origin: Some rigorous resultsMay 10 2019The scaling behaviour of the diffraction intensity near the origin is investigated for (partially) ordered systems, with an emphasis on illustrative, rigorous results. This is an established method to detect and quantify the fluctuation behaviour known ... More

Geometric Estimates in Interpolation by Linear Functions on an Euclidean BallMay 09 2019Let $B_n$ be the Euclidean unit ball in ${\mathbb R}^n$ given by the inequality $\|x\|\leq 1$, $\|x\|:=\left(\sum\limits_{i=1}^n x_i^2\right)^{\frac{1}{2}}$. By $C(B_n)$ we mean the space of continuous functions $f:B_n\to{\mathbb R}$ with the norm $\|f\|_{C(B_n)} ... More