arXiv subjects

Vyron Vellis

Publications and source records attributed to Vyron Vellis.

27 records · Page 2Linked to original sources

arXiv2019

Hölder curves and parameterizations in the Analyst's Traveling Salesman theorem

We investigate the geometry of sets in Euclidean and infinite-dimensional Hilbert spaces. We establish sufficient conditions that ensure a set of points is contained in the image of a $(1/s)$-Hölder continuous map $f:[0,1]\rightarrow l^2$, with $s>1$. Our results are motivated by and generalize the "sufficient half" of the Analyst's Traveling Salesman Theorem, which characterizes subsets of rectifiable curves in $\mathbb{R}^N$ or $l^2$ in terms of a quadratic sum of linear approximation numbers called Jones' beta numbers. The original proof of the Analyst's Traveling Salesman Theorem depends on a well-known metric characterization of rectifiable curves from the 1920s, which is not available for higher-dimensional curves such as Hölder curves. To overcome this obstacle, we reimagine Jones' non-parametric proof and show how to construct parameterizations of the intermediate approximating curves $f_k([0,1])$. We then find conditions in terms of tube approximations that ensure the approximating curves converge to a Hölder curve. As an application, we provide sufficient conditions that guarantee fractional rectifiability of pointwise doubling measures in $\mathbb{R}^N$.

math.CA↗

arXiv2018

Bi-Lipschitz embeddings of Heisenberg submanifolds into Euclidean spaces

The Heisenberg group $\mathbb{H}$ equipped with a sub-Riemannian metric is one of the most well known examples of a doubling metric space which does not admit a bi-Lipschitz embedding into any Euclidean space. In this paper we investigate which \textit{subsets} of $\mathbb{H}$ bi-Lipschitz embed into Euclidean spaces. We show that there exists a universal constant $L>0$ such that lines $L$-bi-Lipschitz embed into $\mathbb{R}^3$ and planes $L$-bi-Lipschitz embed into $\mathbb{R}^4$. Moreover, $C^{1,1}$ $2$-manifolds without characteristic points as well as all $C^{1,1}$ $1$-manifolds locally $L$-bi-Lipschitz embed into $\mathbb{R}^4$ where the constant $L$ is again universal. We also consider several examples of compact surfaces with characteristic points and we prove, for example, that Korányi spheres bi-Lipschitz embed into $\mathbb{R}^4$ with a uniform constant. Finally, we show that there exists a compact, porous subset of $\mathbb{H}$ which does not admit a bi-Lipschitz embedding into any Euclidean space.

math.MG↗

arXiv2018

Geometry of measures in real dimensions via Hölder parameterizations

We investigate the influence that $s$-dimensional lower and upper Hausdorff densities have on the geometry of a Radon measure in $\mathbb{R}^n$ when $s$ is a real number between $0$ and $n$. This topic in geometric measure theory has been extensively studied when $s$ is an integer. In this paper, we focus on the non-integer case, building upon a series of papers on $s$-sets by Martín and Mattila from 1988 to 2000. When $0<s<1$, we prove that measures with almost everywhere positive lower density and finite upper density are carried by countably many bi-Lipschitz curves. When $1\leq s<n$, we identify conditions on the lower density that ensure the measure is either carried by or singular to $(1/s)$-Hölder curves. The latter results extend part of the recent work of Badger and Schul, which examined the case $s=1$ (Lipschitz curves) in depth. Of further interest, we introduce Hölder and bi-Lipschitz parameterization theorems for Euclidean sets with "small" Assouad dimension.

math.CA↗

arXiv2016

Quasiconformal non-parametrizability of almost smooth spheres

We show that, for each $n\ge 3$, there exists a smooth Riemannian metric $g$ on a punctured sphere $\mathbb{S}^n\setminus \{x_0\}$ for which the associated length metric extends to a length metric $d$ of $\mathbb{S}^n$ with the following properties: the metric sphere $(\mathbb{S}^n,d)$ is Ahlfors $n$-regular and linearly locally contractible but there is no quasiconformal homeomorphism between $(\mathbb{S}^n,d)$ and the standard Euclidean sphere $\mathbb{S}^n$.

math.MG↗

arXiv2016

Quasisymmetric extension on the real line

We give a geometric characterization of the sets $E\subset \mathbb{R}$ that satisfy the following property: every quasisymmetric embedding $f: E \to \mathbb{R}^n$ extends to a quasisymmetric embedding $f:\mathbb{R}\to\mathbb{R}^N$ for some $N\geq n$.

math.MG↗

arXiv2016

Bi-Lipschitz embedding of the generalized Grushin plane in Euclidean spaces

We show that, for all $α\geq 0$, the generalized Grushin plane $\mathbb{G}_α$ is bi-Lipschitz homeomorphic to a $2$-dimensional quasiplane in the Euclidean space $\mathbb{R}^{[α]+2}$, where $[α]$ is the integer part of $α$. The target dimension is sharp. This generalizes a recent result of Wu.

math.MG↗

arXiv2015

Quasisymmetric spheres over Jordan domains

Let $Ω$ be a planar Jordan domain. We consider double-dome-like surfaces $Σ$ defined by graphs of functions of $dist( \cdot ,\partial Ω)$ over $Ω$. The goal is to find the right conditions on the geometry of the base $Ω$ and the growth of the height so that $Σ$ is a quasisphere, or quasisymmetric to $\mathbb{S}^2$. An internal uniform chord-arc condition on the constant distance sets to $\partial Ω$, coupled with a mild growth condition on the height, gives a close-to-sharp answer. Our method also produces new examples of quasispheres in $\mathbb{R}^n$, for any $n\ge 3$.

math.MG↗

arXiv2014

Weak chord-arc curves and double-dome quasisymmetric spheres

Let $Ω$ be a planar Jordan domain and $α>0$. We consider double-dome-like surfaces $Σ(Ω,t^α)$ over $\overlineΩ$ where the height of the surface over any point $x\in\overlineΩ$ equals $\text{dist}(x,\partialΩ)^α$. We identify the necessary and sufficient conditions in terms of $Ω$ and $α$ so that these surfaces are quasisymmetric to $\mathbb{S}^2$ and we show that $Σ(Ω,t^α)$ is quasisymmetric to the unit sphere $\mathbb{S}^2$ if and only if it is linearly locally connected and Ahlfors $2$-regular.

math.MG↗

arXiv2013

Sets of constant distance from a Jordan curve

We study the $ε$-level sets of the signed distance function to a planar Jordan curve $Γ$, and ask what properties of $Γ$ ensure that the $ε$-level sets are Jordan curves, or uniform quasicircles, or uniform chord-arc curves for all sufficiently small $ε$. Sufficient conditions are given in term of a scaled invariant parameter for measuring the local deviation of subarcs from their chords. The chordal conditions given are sharp.

math.MG↗