Smallest area shape that covers all unit length curve - MathOverflow

Smallest area shape that covers all unit length curve

2010-07-19T12:27:11Z

On a euclidean plane, what is the minimal area shape S, such that for every unit length curve, a translation and a rotation of S can cover the curve.

What are the bounds of the shape's area if this is a open problem?

When I asked this problem few years ago, someone told me it's open. I don't know if this is still open and I can't find any reference on it.

I don't even know what branch of mathematics it falls under. so I can't even tag this question.

Answer by Joseph O'Rourke for Smallest area shape that covers all unit length curve

2010-07-19T12:43:28Z

P.A.P. Moran proved in 1946, in "On a Problem of S. Ulam" [J. London Math. Soc. 1946 s1-21: 175-179] this theorem:

If $C$ is a curve of unit length in the plane, and $|K$| is the area of its smallest convex cover $K$, then $|K| \le 1/(2\pi)$, and this is the best possible result, since this limit is attained for a semicircle of unit length.

This may not answer your questions entirely, but perhaps it can seed your search.

Answer by Andrey Rekalo for Smallest area shape that covers all unit length curve

2010-07-19T12:45:32Z

Whereas I don't know of any recent progress in this problem, let me mention one result for closed curves.

Theorem. A closed plane curve of length $L$ and curvature bounded by $K$ can be contained inside a circle of radius $L/4 - (\pi - 2)/2K$.

This was proved in 1974 by H.H. Johnson (link 1) who used calculus of variations methods. A geometric proof was given a bit later by Chakerian, Johnson and Vogt (link 2).

Edit. Apparently the problem is still open. Here's an article (arXiv link), which contains a survey of some known results as of 2009. From the Introduction:

In 1966, Leo Moser asked for the region of smallest area which can accommodate every planar arc of length one. The problem is known as “Moser’s worm problem” and is a variation of universal cover problems. In Moser’s problem, a cover is a set which contains a copy of any rectifiable planar arc of unit length, and is usually assumed to be convex. Such a minimal cover is known to have area between 0.2194 and 0.2738. However, the original problem remains unsolved.

Answer by yatima2975 for Smallest area shape that covers all unit length curve

2010-07-19T12:51:14Z

There's a chapter in Ian Stewart's 'Game, Set and Math' that covers this problem in a very accessible way (in the guise of a blanket for a worm, IIRC). I'm pretty sure that Joseph's semicircle is the right answer, but it's been ages since I've read that book.

Answer by Thorny for Smallest area shape that covers all unit length curve

2010-07-19T13:13:22Z

Reportedly R. Norwood, G. Poole, M. Laidacker: The Worm Problem of Leo Moser, Discrete & Computational Geometry 7 (1992), 153-162. has an example of area $\sqrt{3}/12+\pi/24$ (a 60 degree sector of a circle with two triangular "wings"), and this was the best known in 1999.

Answer by A B for Smallest area shape that covers all unit length curve

2010-07-19T18:43:00Z

The lower bound (initially provided by Khandawit and Sriwasdi) was improved in 2009 by Dimitrios Pagonakis. Although the paper has not been put on the arxiv yet (I understand that the author has plans to do this), the bound was improved from 0.227498 to 0.231999.

Cf. Page 8/12 in http://www-math.mit.edu/news/summer/2009RSIAbstracts.pdf

Answer by Dimitrios Pagonakis for Smallest area shape that covers all unit length curve

2011-02-06T22:02:27Z

It took some time but now is on arxiv: http://arxiv.org/abs/1101.5638