Related papers: Between Shapes, Using the Hausdorff Distance

We obtain new lower bounds on the Hausdorff dimension of distance sets and pinned distance sets of planar Borel sets of dimension slightly larger than $1$, improving recent estimates of Keleti and Shmerkin, and of Liu in this regime. In…

Let $s \in [0,1]$. We show that a Borel set $N \subset \mathbb{R}^{2}$ whose every point is linearly accessible by an $s$-dimensional family of lines has Hausdorff dimension at most $2 - s$.

The aperture angle alpha(x, Q) of a point x not in Q in the plane with respect to a convex polygon Q is the angle of the smallest cone with apex x that contains Q. The aperture angle approximation error of a compact convex set C in the…

$\theta$ intermediate dimensions are a continuous family of dimensions that interpolate between Hausdorff and Box dimensions of fractal sets. In this paper we study the problem of the relationship between the dimension of a set…

We discuss asymptotics for large random planar maps under the assumption that the distribution of the degree of a typical face is in the domain of attraction of a stable distribution with index $\alpha\in(1,2)$. When the number $n$ of…

This paper contains new results on two classical topics in fractal geometry: projections, and intersections with affine planes. To keep the notation of the abstract simple, we restrict the discussion to the planar cases of our theorems. Our…

We explore convex shapes $S$ in the Euclidean plane which have the following property: there is a circle $C$ such that the angle between the two tangents from any point of $C$ to $S$ is constant equal to $\alpha$. A dynamical formulation…

Let $M$ be a simple 3-manifold such that one component of $\partial M$, say $F$, has genus at least two. For a slope $\alpha$ on $F$, we denote by $M(\alpha)$ the manifold obtained by attaching a 2-handle to $M$ along a regular neighborhood…

We introduce an equivalence relation on $W^{s,p}({\mathbb S}^N;{\mathbb S}^N)$ involving the topological degree, and we evaluate the distances (in the usual sense and in the Hausdorff sense) between the equivalence classes. In some special…

A classical theorem due to Mattila (see \cite{Mat84}; see also \cite{M95}, Chapter 13) says that if $A,B \subset {\Bbb R}^d$ of Hausdorff dimension $s_A, s_B$, respectively, with $s_A+s_B \ge d$, $s_B>\frac{d+1}{2}$ and $dim_{{\mathcal…

When matching parts of a surface to its whole, a fundamental question arises: Which points should be included in the matching process? The issue is intensified when using isometry to measure similarity, as it requires the validation of…

Given $n$ distinct points $\mathbf{x}_1, \ldots, \mathbf{x}_n$ in $\mathbb{R}^d$, let $K$ denote their convex hull, which we assume to be $d$-dimensional, and $B = \partial K $ its $(d-1)$-dimensional boundary. We construct an explicit…

A characterization of the proximal normal cone is obtained and a separation theorem for convex subsets of Riemannian manifolds is established. Moreover, the convexity of the distance function $d_S$ for a convex subset $S$ in the cases where…

We give a full description for the shape of the classical (2+1)\Dim Solid-On-Solid model above a wall, introduced by Temperley (1952). On an $L\times L$ box at a large inverse-temperature $\beta$ the height of most sites concentrates on a…

In this paper, we introduce the mean $\Psi$-intermediate dimension which has a value between the mean Hausdorff dimension and the metric mean dimension, and prove the equivalent definition of the mean Hausdorff dimension and the metric mean…

Consider a set P of N random points on the unit sphere of dimension $d-1$, and the symmetrized set S = P union (-P). The halving polyhedron of S is defined as the convex hull of the set of centroids of N distinct points in S. We prove that…

We introduce an equivalence relation on the space $W^{1,1}(\Omega;{\mathbb S}^1)$ which classifies maps according to their "topological singularities". We establish sharp bounds for the distances (in the usual sense and in the Hausdorff…

This document offers a concise introduction to the mathematical theory and practical application of the Hausdorff Measure and Dimension. The primary objective is to clarify and rigorously detail the two most common methods used for…

We determine the extent to which certain classes of fractionally `smooth' continuous mappings between metric spaces distort various dimensions, including the Hausdorff, upper Minkowski (box-counting), and upper intermediate dimensions. Our…

We provide general upper and lower bounds for the Gromov-Hausdorff distance $d_{\mathrm{GH}}(\mathbb{S}^m,\mathbb{S}^n)$ between spheres $\mathbb{S}^m$ and $\mathbb{S}^n$ (endowed with the round metric) for $0\leq m< n\leq \infty$. Some of…