A Lower Bound for Lebesgue's Universal Cover Problem

Braß, Peter; Sharifi, Mehrbod

doi:10.1142/s0218195905001828

Cited by 17 publications

(16 citation statements)

References 3 publications

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“…In recent years computational methods have increasingly been employed to attack geometric questions. For example, Brass and his student Sharifi [3] used a grid search numerical method to improve the known lower bound for the Lebesgue Universal Cover problem. In this article we employ numerical convex optimization to reduce the known upper bound for α 2 by nearly 1.7 per cent.…”

Section: A Smaller Covermentioning

confidence: 99%

A smaller cover for closed unit curves

Wichiramala¹

2018

MMN

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Forty years ago Schaer and Wetzel showed that a 1 π × 1 2π √ π 2 − 4 rectangle, whose area is about 0.122 74, is the smallest rectangle that is a cover for the family of all closed unit arcs. More recently Füredi and Wetzel showed that one corner of this rectangle can be clipped to form a pentagonal cover having area 0.11224 for this family of curves.Here we show that then the opposite corner can be clipped to form a hexagonal cover of area less than 0.11023 for this same family. This irregular hexagon is the smallest cover currently known for this family of arcs.keywords: covering of unit arcs, covering by convex sets, worm problem MSC code: 52A38

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Section: A Smaller Covermentioning

confidence: 99%

A smaller cover for closed unit curves

Wichiramala¹

2018

MMN

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“…By Lemma 1 both connected components, denoted by C 1 and C 2 , are of diameter at most 1, so Jung's theorem [15,28] Note that we used a bit wastefully the Jung enclosing balls. The universal cover problem, first stated in a personal communication of Lebesgue in 1914, asks for the minimum area A of a convex set U containing a congruent copy of any planar set of diameter 1, see [8]. For the currently best known bounds 0.832 ≤ A ≤ 0.844 and generalizations to higher dimensions we refer the interested reader to [7,Section 11.4].…”

Section: Bounds For L D (N)mentioning

confidence: 99%

Open Sets Avoiding Integral Distances

Kurz

Mishkin

2013

Discrete Comput Geom

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We study open point sets in Euclidean spaces R d without a pair of points an integral distance apart. By a result of Furstenberg, Katznelson, and Weiss such sets must be of Lebesgue upper density zero. We are interested in how large such sets can be in d-dimensional volume. We determine the exact values for the maximum volumes of the sets in terms of the number of their connected components and dimension. Here techniques from diophantine approximation, algebra and the theory of convex bodies come into play. Our problem can be viewed as a counterpart to known problems on sets with pairwise rational or integral distances. This reveals interesting links between discrete geometry, topology, and measure theory.

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“…Lebesgue's problem was first studied by Pál [8], who found that the area of a smallest universal cover is at least 0.8257 and at most 0.8454. Both bounds were improved by several authors, the current best upper bound is around 0.844 [3], the best lower bound is around 0.832 [5], so the problem is still open.…”

Section: Introductionmentioning

confidence: 99%