• Corpus ID: 119140340

A New Slant on Lebesgue’s Universal Covering Problem

  title={A New Slant on Lebesgue’s Universal Covering Problem},
  author={Philip Gibbs},
  journal={arXiv: Metric Geometry},
  • P. Gibbs
  • Published 29 January 2014
  • Mathematics
  • arXiv: Metric Geometry
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… 
An Upper Bound for Lebesgue’s Covering Problem
A covering problem posed by Henri Lebesgue in 1914 seeks to find the convex shape of smallest area that contains a subset congruent to any point set of unit diameter in the Euclidean plane. Methods
Errata and Addenda to Mathematical Constants
We humbly and briefly offer corrections and supplements to Mathematical Constants (2003) and Mathematical Constants II (2019), both published by Cambridge University Press. Comments are always
Open Peer Review to Save the World
Humanity faces many dangers from climate change and wars to asteroid impacts that could harm our future. Often logical reasoning does not seem to play a strong part in discussions on such subjects


A Lower Bound for Lebesgue's Universal Cover Problem
Any convex set that contains a congruent copy of any set of diameter one (universal cover) has area at least 0.832, considerably improves the lower bound for Lebesgue's universal cover problem, using a combination of computer search and geometric bounds.
Small universal covers for sets of unit diameter
The current smallest convex universal cover for sets of unit diameter is described. This reduction of Sprague's cover is by 4 · 10-11 and results in an asymmetrical cover. Another small universal
Research problems in discrete geometry
This chapter discusses 100 Research Problems in Discrete Geometry from the Facsimile edition of the World Classics in Mathematics Series, vol.
Generalized breadths, circular Cantor sets, and the least area UCC
  • G. Elekes
  • Mathematics
    Discret. Comput. Geom.
  • 1994
We develop a technique suitable for determining the minimal area convex set that can cover certain prescribed regular polygons. As a side effect we improve the well-known “circle-and-triangle” lower
Geometry and Convexity: A Study in Mathematical Methods
Helps students see mathematics as an organic whole by focusing on the geometric while presenting viewpoints and methods that require a general understanding and unification of previous mathematical
Über ein elementares variationsproblem
  • Mat. Tidsskrift
  • 1936
Pach : “ Research problems in discrete geometry
  • 2005
Towards the minimal universal cover
  • Normat
  • 1981
A smaller universal cover for sets of unit diameter
  • C. R. Math. Acad. Sci
  • 1980
On Universal Covering Sets and Translation Covers in the Plane
  • James Cook Mathematical Notes,
  • 1980