Convex and discrete geometry Convex geometric analysis and geometric inequalities Problem

Moser's worm problem

Builds on Arc length Isometry Plane curve

Problem

Call a region $R$ in the plane a worm cover if it has the following property. For every plane curve $\gamma$ of length $1$ , some isometry $g$ of the plane satisfies $g(\gamma) \subseteq R$ . Find the smallest possible area of a worm cover.

Remarks

The problem is open; most work concerns convex worm covers. A disc of radius $1/2$ is such a region, so the answer is at most $\pi / 4$ . Indeed, let $m$ be the point of $\gamma$ at arc length $1/2$ from an end. Every point of $\gamma$ is joined to $m$ by an arc of length at most $1/2$ , so it lies within distance $1/2$ of $m$ . Much smaller convex worm covers are known.

Reading path

To understand this page from the ground up, read these 4 pages in order. The foundations come first.

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Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.

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Arc length concept
Isometry concept
Plane curve concept
Moser's worm problem problem

Tag	`DTVFW3QMNW`
URI	`m2://DTVFW3QMNW`
Kind	problem
Writing	W0 stub
MSC	52C15, 52A40
Status	This copy of the site has no build data. Run `m2 status DTVFW3QMNW`.
About	Plane curve
Uses	Arc length, Isometry, Plane curve
Same as	Wikidata Q15904307 Wikipedia Moser's worm problem

Moser's worm problem

Problem

Remarks

Reading path

Further reading

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