Related papers: Remarks on WDC sets

We prove a version of Bourgain's projection theorem for parametrized families of $C^2$ maps, that refines the original statement even in the linear case. As one application, we show that if $A$ is a Borel set of Hausdorff dimension close to…

The purpose of this paper is to give a survey on the notions of distance between subsets either of a metric space or of a measure space, including definitions, a classification, and a discussion of the best-known distance functions, which…

We consider the reverse math strength of the statement $\mathsf{C\text-DM}$:"Every completely determined Borel set is measurable." Over $\mathsf{WWKL}_0$, we obtain the following results analogous to the previously studied category case.…

If $(X,d)$ is a Polish metric space of dimension $0$, then by Wadge's lemma, no more than two Borel subsets of $X$ can be incomparable with respect to continuous reducibility. In contrast, our main result shows that for any metric space…

We consider a definition of a weakly convex set which is a generalization of the notion of a weakly convex set in the sense of Vial and a proximally smooth set in the sense of Clarke, from the case of the Hilbert space to a class of Banach…

For a large class of self-similar sets F in R^d analogues of the higher order mean curvatures of differentiable submanifolds are introduced, in particular, the fractal Gauss-type curvature. They are shown to be the densities of associated…

We investigate the size of the distance set determined by two subsets of finite dimensional vector spaces over finite fields. A lower bound of the size is given explicitly in terms of cardinalities of the two subsets. As a result, we…

An achievement set of a series is a set of all its subsums. We study the properties of achievement sets of conditionally convergent series in finite dimensional spaces. The purpose of the paper is to answer some of the open problems…

A subset of a metric space is a k-distance set if there are exactly k non-zero distances occuring between points. We conjecture that a k-distance set in a d-dimensional Banach space (or Minkowski space), contains at most (k+1)^d points,…

In this article, we study the (d-1)-volume and the covering numbers of the medial axis of a compact set of the Euclidean d-space. In general, this volume is infinite; however, the (d-1)-volume and covering numbers of a filtered medial axis…

In this work we characterize the sets $E\subset X$ for which there is some $\alpha>0$ such that the function $d(\cdot,E)^{-\alpha}$ belongs to the Muckenhoupt class $A_1(X,d,\mu)$, where $(X,d,\mu)$ is a space of homogeneous type, extending…

It is known that for $X$ a nowhere locally compact metric space, the set of bounded continuous, nowhere locally uniformly continuous real-valued functions on $X$ contains a dense $G_\delta$ set in the space $C_b(X)$ of all bounded…

This paper discusses the properties of the spaces of fuzzy sets in a metric space with $L_p$-type $d_p$ metrics, $p\geq 1$. The $d_p$ metrics are well-defined if and only if the corresponding Haudorff distance functions are measurable. In…

Let (X,d) be a metric space and (\Omega, d) a compact subspace of X which supports a non-atomic finite measure m. We consider `natural' classes of badly approximable subsets of \Omega. Loosely speaking, these consist of points in \Omega…

Self-projective sets are natural fractal sets which describe the action of a semigroup of matrices on projective space. In recent years there has been growing interest in studying the dimension theory of self-projective sets, as well as…

The convexity of a set can be generalized to the two weaker notions of reach and $r$-convexity; both describe the regularity of a set's boundary. For any compact subset of $\mathbb{R}^d$, we provide methods for computing upper bounds on…

Dobi\'nski set $\mathcal{D}$ is an exceptional set for a certain infinite product identity, whose points are characterized as having exceedingly good approximations by dyadic rationals. We study the Hausdorff dimension and logarithmic…

We characterize the subsets $E$ of a metric space $X$ with doubling measure whose distance function to some negative power $\textrm{dist}(\cdot,E)^{-\alpha}$ belongs to the Muckenhoupt $A_1$ class of weights in $X$. To this end, we…

We construct a compact metric space that has any other compact metric space as a tangent, with respect to the Gromov-Hausdorff distance, at all points. Furthermore, we give examples of compact sets in the Euclidean unit cube, that have…

A $W^{1,p}$-metric on an $n$-dimensional closed Riemannian manifold naturally induces a distance function, provided $p$ is sufficiently close to $n$. If a sequence of metrics $g_k$ converges in $W^{1,p}$ to a limit metric $g$, then the…