Related papers: A note on topological dimension, Hausdorff measure…

We review the motivation and fundamental properties of the Hausdorff dimension of metric spaces and illustrate this with a number of examples, some of which are expected and well-known. We also give examples where the Hausdorff dimension…

We introduce a new notion of a harmonic measure for a $d$-dimensional set in $\R^n$ with $d<n-1$, that is, when the codimension is strictly bigger than 1. Our measure is associated to a degenerate elliptic PDE, it gives rise to a…

Given a compact $n$-dimensional immersed Riemannian manifold $M^n$ in some Euclidean space we prove that if the Hausdorff dimension of the singular set of the Gauss map is small, then $M^n$ is homeomorphic to the sphere $S^n$. Also, we…

This paper is devoted to the proof of two related results. The first one asserts that if $\mu$ is a Radon measure in $\mathbb R^d$ satisfying $$\limsup_{r\to 0} \frac{\mu(B(x,r))}{r}>0\quad \text{ and }\quad…

The deck, $\mathcal{D}(X)$, of a topological space $X$ is the set $\mathcal{D}(X)=\{[X \setminus \{x\}]\colon x \in X\}$, where $[Y]$ denotes the homeomorphism class of $Y$. A space $X$ is (topologically) reconstructible if whenever…

This article studies typical 1-Lipschitz images of $n$-rectifiable metric spaces $E$ into $\mathbb{R}^m$ for $m\geq n$. For example, if $E\subset \mathbb{R}^k$, we show that the Jacobian of such a typical 1-Lipschitz map equals 1…

I prove that the visible parts of a compact set in $\mathbb{R}^{n}$, $n \geq 2$, have Hausdorff dimension at most $n - \tfrac{1}{50n}$ from almost every direction.

We give a sufficient condition for a Borel subset $E\subset X$ of a complete metric space with $\mathcal{H}^n(E)<\infty$ to be $n$-rectifiable. This condition involves a decomposition of $E$ into rectifiable curves known as an Alberti…

Methods for measuring convexity defects of compacts in R^n abound. However, none of the those measures seems to take into account continuity. Continuity in convexity measure is essential for optimization, stability analysis, global…

We characterise rectifiable subsets of a complete metric space $X$ in terms of local approximation, with respect to the Gromov--Hausdorff distance, by an $n$-dimensional Banach space. In fact, if $E\subset X$ with $\mathcal{H}^n(E)<\infty$…

Hurewicz' characterized the dimension of separable metrizable spaces by means of finite-to-one maps. We investigate whether this characterization also holds in the class of compact F-spaces of weight c. Our main result is that, assuming the…

In the present paper we sketch the proof of the fact that for any open connected set $\Omega\subset\mathbb{R}^{n+1}$, $n\geq 1$, and any $E\subset \partial \Omega$ with $0<\mathcal{H}^n(E)<\infty$, absolute continuity of the harmonic…

Let $(X,d)$ be an $n$-dimensional Alexandrov space whose Hausdorff measure $\mathcal{H}^n$ satisfies a condition giving the metric measure space $(X,d,\mathcal{H}^n)$ a notion of having nonnegative Ricci curvature. We examine the influence…

Given a compact set $E \subset \mathbb{R}^{d - 1}$, $d \geq 1$, write $K_{E} := [0,1] \times E \subset \mathbb{R}^{d}$. A theorem of C. Bishop and J. Tyson states that any set of the form $K_{E}$ is minimal for conformal dimension: if…

When one considers the collection $\mathcal{H}(\mathbb{R}^n)$ of all compact subsets of $\mathbb{R}^n$ and equip it with a topology, many questions can be asked about the topological space one ends up with. This is an example of a…

In this work, we are interested in characterizing typical (generic) dimensional properties of invariant measures associated with the full-shift system, $T$, in a product space whose alphabet is a perfect and separable metric space (thus,…

A homemorphism between domains in $\mathbb R^n$, $n\ge 2$ is quasiconformal, with its intricate analytic and geometric consequences, if the (pointwise) linear dilatation -- a purely metric quantity -- is uniformly bounded. Gehring proved…

In the present paper we prove that for any open connected set $\Omega\subset{\mathbb R}^{n+1}$, $n\geq 1$, and any $E\subset \partial\Omega$ with $0<{\mathcal H}^n(E)<\infty$ absolute continuity of the harmonic measure $\omega$ with respect…

In this short note, for countably infinite amenable group actions, we provide topological proofs for the following results: Bowen topological entropy (dimensional entropy) of the whole space equals the usual topological entropy along…

Federer'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…