Lower Bound for Convex Hull Area and Universal Cover Problems (original) (raw)
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Lower Bounds Of Areas Of Convex Covers For Closed Unit Arcs
2020
Moser's worm problem is the unsolved problem in geometry which asks for the minimal area of a region S on the plane which can cover all curves of unit length, assuming that curves may be rotated and translated to fit inside the region. This thesis studies a version of this problem when region S is convex and unit curves to be covered are closed. For example, region S should be able to cover a circle of length 1, a square of side length 1/4, a line interval of length 1/2, and so on. An example of such cover S is the circle of diameter 1, whose area is about 0:7854, but the problem is to find S with minimal area. Recently, Wichiramala constructed a hexagon with this property and area about 0.11023, and this is the current record. On the other hand, it is known that the area of S cannot be less than 0.096694.In this work, we improve the lower bound for area of convex cover S for closed unit arcs from 0.096694 to 0.0975 and then to 0.1 by finding the smallest areas of convex hulls o...
A convex cover for closed unit curves has area at least 0.0975
2020
We combine geometric methods with numerical box search algorithm to show that the minimal area of a convex set on the plane which can cover every closed plane curve of unit length is at least 0.0975. This improves the best previous lower bound of 0.096694. In fact, we show that the minimal area of convex hull of circle, equilateral triangle, and rectangle of perimeter 1 is between 0.0975 and 0.09763.
Universal convex covering problems under translation and discrete rotations
arXiv (Cornell University), 2022
We consider the smallest-area universal covering of planar objects of perimeter 2 (or equivalently closed curves of length 2) allowing translation and discrete rotations. In particular, we show that the solution is an equilateral triangle of height 1 when translation and discrete rotation of π are allowed. Our proof is purely geometric and elementary. We also give convex coverings of closed curves of length 2 under translation and discrete rotations of multiples of π/2 and 2π/3. We show a minimality of the covering for discrete rotation of multiples of π/2, which is an equilateral triangle of height smaller than 1, and conjecture that the covering is the smallest-area convex covering. Finally, we give the smallest-area convex coverings of all unit segments under translation and discrete rotations 2π/k for all integers k ≥ 3.
A smaller cover for closed unit curves
Miskolc Mathematical Notes
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.
An Improved Upper Bound for Leo Moser's Worm Problem
Discrete and Computational Geometry, 2003
A worm ω is a continuous rectifiable arc of unit length in the Cartesian plane. Let W denote the class of all worms. A planar region C is called a cover for W if it contains a copy of every worm in W. That is, C will cover or contain any member ω of W after an appropriate translation and/or rotation of ω is completed (no reflections). The open problem of determining a cover C of smallest area is attributed to Leo Moser [7], [8]. This paper reduces the smallest known upper bound for this area from 0.275237 [10] to 0.260437.
Covering n-Segment Unit Arcs Is Not Sufficient
In 1974 Gerriets and Poole conjectured for n = 3 that a convex set in the plane which contains a congruent copy of every n-segment polygonal arc of unit length must be a cover for the family of all unit arcs. We disprove this general conjecture by describing for each positive integer n a convex region Rn that contains a congruent copy of every n-segment unit arc but not a congruent copy of every unit arc.
Discrete & Computational Geometry, 1992
One of Leo Moser's geometry problems is referred to as the Worm Problem [10]: "What is the (convex) region of smallest area which will accommodate (or cover) every planar arc of length 1?" For example, it is easy to show that the circular disk with diameter 1 will cover every planar arc of length 1. The area of the disk is approximately 0.78539. Here we show that a solution to the Worm Problem of Moser is a region with area less than 0.27524.
Capacitated Covering Problems in Geometric Spaces
Discrete and Computational Geometry, 2019
In this article, we consider the following capacitated covering problem. We are given a set P of n points and a set B of balls from some metric space, and a positive integer U that represents the capacity of each of the balls in B. We would like to compute a subset B ⊆ B of balls and assign each point in P to some ball in B that contains it, such that the number of points assigned to any ball is at most U. The objective function that we would like to minimize is the cardinality of B. We consider this problem in arbitrary metric spaces as well as Euclidean spaces of constant dimension. In the metric setting, even the uncapacitated version of the problem is hard to approximate to within a logarithmic factor. In the Euclidean setting, the best known approximation guarantee in dimensions 3 and higher is logarithmic in the number of points. Thus we focus on obtaining "bi-criteria" approximations. In particular, we are allowed to expand the balls in our solution by some factor, but optimal solutions do not have that flexibility. Our main result is that allowing constant factor expansion of the input balls suffices to obtain constant approximations for this problem. In fact, in the Euclidean setting, only (1 +) factor expansion is sufficient for any > 0, with the approximation factor being a polynomial in 1/. We obtain these results using a unified scheme for rounding the natural LP relaxation; this scheme may be useful for other capacitated covering problems. We also complement these bi-criteria approximations by obtaining hardness of approximation results that shed light on our understanding of these problems. 2012 ACM Subject Classification Theory of computation → Packing and covering problems, Theory of computation → Rounding techniques, Theory of computation → Computational geometry, Mathematics of computing → Approximation algorithms
On the geometric dilation of curves and point sets
2004
Let G be an embedded planar graph whose edges are curves. The detour between two points u and v (on edges or vertices) of G is the ratio between the shortest path in G between u and v and their Euclidean distance. The maximum detour over all pairs of points is called the geometric dilation δ(G). Ebbers-Baumann, Grüne and Klein have recently shown that every finite point set is contained in a planar graph whose geometric dilation is at most 1.678, and some point sets require graphs with dilation δ ≥ π/2 ≈ 1.57. We prove a stronger lower bound δ ≥ (1 + 10 −11 )π/2 by relating graphs with small dilation to a problem of packing and covering the plane by circular disks.
On the best estimate for perimeters of plane sets with the angle property
Geometriae Dedicata, 1996
We investigate the problem of finding the maximum length of perimeters of plane sets with fixed diameter d, such that every point of the boundary of the set is a vertex of an open angle of opening c~ which does not intersect the set. First we consider plane curves which satisfy such angle property in a finite number of directions, and among them we find the one of maximum length. Then we prove that the perimeter of any plane set with the angle property is less than or equal to 7rd(sin a/2)-2; this is the best estimate when 7r/2 < c~ < 7r.