A convex cover for closed unit curves has area at least 0.1

Abstract

We improve a lower bound for the smallest area of convex covers for closed unit curves from 0.0975 to 0.1, which makes it substantially closer to the current best upper bound 0.11023. We did this by considering the minimal area of convex hull of circle, line of length $\frac{1}{2}$, and rectangle with side $0.1727\times 0.3273$. By using geometric methods and the Box-search algorithm, we proved that this area is at least 0.1. We give informal numerical evidence that the obtained lower bound is close to the limit of current techniques, and substantially new idea is required to go significantly beyond 0.10044.

Introduction

Moser’s worm problem is one of many unsolved problems in geometry posed by Leo Moser [1] in 1966 (see [2] for a new version). It asks about the smallest area of region $S$ in the plane which can be rotated and translated to cover every curve of length $1$ (unit curve). While this problem is still open, lower and upper bounds for this smallest area are available in the literature. The current best upper bound 0.260437 was established by Norwood and Poole [3]. For the lower bound, it is known only that the area of $S$ is strictly positive [4].

This problem can be modified by restricting region $S$ (e.g. triangle, rectangle, convex, non-convex, etc.) or restricting unit curves (e.g. closed, polygonal, etc.). Numerous problems in the worm family have been solved, see Wetzel [5] for a survey.

If we require $S$ to be convex, and cover arbitrary unit curves, the existence of the solution is shown by Laidacker and Poole [6] using Blaschke Selection Theorem. In 2019, Panraksa and Wichiramala [7] showed that the Wetzel’s sector is a cover for unit curves which has an area of $\pi ∕12\approx 0.2618$. This cover is the current upper bound for this problem, whereas the current lower bound 0.232239 was found by Khandhawit, Sriswasdi and Pagonakis [8] in 2013.

The problem of finding smallest convex region $S$ to cover all (not necessarily unit) curves of diameter one was posed by Henri Lebesgue in 1914 and is known as Lebesgue’s universal covering problem. In this case, the current best lower and upper bounds are 0.832 and 0.8440936 due to Peter Brass and Mehrbod Sharifi [9] and Philip Gibbs [10], respectively.

In this paper, we consider the problem of covering closed unit curves by a convex region of the smallest area.

In 1957, H.G. Eggleston [11] showed that the equilateral triangle of side $\frac{\sqrt{3}}{\pi }$ is the smallest triangle which can cover all closed unit curves and its area is $\frac{3\sqrt{3}}{4{\pi }^2}\approx 0.13162$. Twenty-five years later, the smallest rectangle with lengths $\frac{1}{\pi }$ and $\sqrt{\frac{1}{4}-\frac{1}{{\pi }^2}}$ is found by Shaer and Wetzel [12] and its area is about 0.12274.

Füredi and Wetzel [13] decreased this area to about 0.11222 by clipping one corner of this rectangle. Also, they modified the resulted pentagon to a curvilinear hexagon which has an area of less than 0.11213. In 2018, Wichiramala [14] showed that a corner of the pentagon can be clipped to obtain a hexagonal cover with area less than 0.11023, which is the best currently known upper bound.

Our work focuses on a lower bound for this problem. In 1973, Chakerian and Klamkin [15] applied Fáry and Rédei’s theorem [16] to find the first lower bound 0.0963275 by considering the convex hull of circle and line segment. Further, Füredi and Wetzel [13] showed that the minimum area of convex hull of circle and the $0.0261682\times 0.4738318$ rectangle has area 0.0966675. Moreover, they replaced this rectangle by curvilinear rectangle and give a new lower bound about 0.096694.

To improve this bound, one may consider the minimal convex hull of three given closed curves. However, in this case Fáry and Rédei’s theorem [16] cannot be applied to find smallest area analytically, which complicates the analysis and call for the mix of geometric and numerical methods. Som-am [17] applied Brass grid method [9] to prove that the area of convex hull of circle, line segment, and equilateral triangle is at least 0.096905. In 2019, Grechuk and Som-am [18] used the Box search method to show that the minimal-area convex hull of a rectangle of perimeter $1$, circle with perimeter $1$, and the equilateral triangle with perimeter $1$, is greater than 0.0975. Thus, if we denote $\alpha$ to be the area of smallest cover of closed unit curves, then

$$0.0975\le \alpha \le 0.11023.$$

In this paper, we use geometric methods combined with the Box-search algorithm to prove that the area of convex cover for the line segment, circle, and certain rectangle of perimeter $1$ is at least 0.1.

Theorem 1.1

The area of convex cover $S$ for circle of circumference $1$, line of length $1∕2$, and rectangle of size $0.1727\times 0.3273$ is at least 0.1.

By Theorem 1.1, we have

$$0.1<\alpha ,$$

which improves the previous bound 0.0975. In fact, the true minimal area of $S$ in Theorem 1.1 is about 0.10044, but we have rounded the bound to 0.1 to simplify the proof.

The choice of line segment, circle, and rectangle in Theorem 1.1 was not an arbitrary or lucky one. In fact, we have used a systematic search for three shapes which give the best possible lower bound.

The organization of this paper is as follows. A systematic search for three shapes with maximal possible area of minimal convex hull is presented in Section 2. The remaining sections are devoted to the proof of Theorem 1.1. Section 3 proves some geometric lemmas. Section 4 describes numerical Box-search algorithm and computational results. The proof of Theorem 1.1 is finished in Section 5. We give some conclusions in Section 6.