On universal covers for four-dimensional sets of a given diameter

Lower Bound for Convex Hull Area and Universal Cover Problems

In this paper, we provide a lower bound for an area of the convex hull of points and a rectangle in a plane. We then apply this estimate to establish a lower bound for a universal cover problem. We showed that a convex universal cover for a unit length curve has area at least 0.232239. In addition, we show that a convex universal cover for a unit closed curve has area at least 0.0879873.

关于勒贝格通用覆盖问题的注记

Note on Lebesgue's universal cover problem

该注记研究勒贝格通用覆盖问题，证明三维外接直径1球的正十二面体非通用覆盖，改进三维通用覆盖最小体积界，给出$d$维外接单位球的最大面数中心对称通用覆盖多面体族，推广并改进了相关界。

Generalized Universal Covers of Uniform Spaces

We develop a generalized covering space theory for a class of uniform spaces called coverable spaces. Coverable spaces include all geodesic metric spaces, connected and locally pathwise connected compact topological spaces, in particular Peano continua, as well as more pathological spaces like the topologist's sine curve. Each coverable space has a generalized universal cover with universal and lifting properties. Associated with this generalized universal cover is a functorial uniform space invariant called the deck group, which is related to the classical fundamental group by a natural homom…

A New Slant on Lebesgue's Universal Covering Problem

Lebesgue's universal covering problem is re-examined using computational methods. This leads to conjectures about the nature of the solution which if correct could provide a blueprint for a complete solution. Empirical lower bounds for the minimal area are computed using different hypotheses based on the conjectures. A new upper bound of 0.844113 for the area of the minimal cover is derived improving previous results. This method for determining the bound is suggested by the conjectures and computational observations but is proved independently of them. The key innovation is to modify previous…

Non-existence of the box dimension for dynamically invariant sets

One of the key challenges in the dimension theory of smooth dynamical systems is in establishing whether or not the Hausdorff, lower and upper box dimensions coincide for invariant sets. For sets invariant under conformal dynamics, these three dimensions always coincide. On the other hand, considerable attention has been given to examples of sets invariant under non-conformal dynamics whose Hausdorff and box dimensions do not coincide. These constructions exploit the fact that the Hausdorff and box dimensions quantify size in fundamentally different ways, the former in terms of covers by sets …

关于渐近勒贝格通用覆盖问题

On asymptotic Lebesgue's universal covering problem

本文研究了高维情况下Jung球的渐近最优性，证明了通用覆盖的体积下界。

The Revised and Uniform Fundamental Groups and Universal Covers of Geodesic Spaces

Sormani and Wei proved in 2004 that a compact geodesic space has a categorical universal cover if and only if its covering/critical spectrum is finite. We add to this several equivalent conditions pertaining to the geometry and topology of the revised and uniform fundamental groups. We show that a compact geodesic space X has a universal cover if and only if the following hold: 1) its revised and uniform fundamental groups are finitely presented, or, more generally, countable; 2) its revised fundamental group is discrete as a quotient of the topological fundamental group. In the process, we cl…

On the universal cover and the fundamental group of an $RCD^*(K,N)$-space

The main goal of the paper is to prove the existence of the universal cover for $RCD^*(K,N)$-spaces. This generalizes earlier work of C. Sormani and the second named author on the existence of universal covers for Ricci limit spaces. As a result, we also obtain several structure results on the (revised) fundamental group of such spaces. These are the first topological results for $RCD^{*}(K,N)$-spaces without extra structural-topological assumptions (such as semi-local simple connectedness).