Kakeya needle problem

Problem

A needle is a line segment of length $1$ in the plane $\R^2$ . Find the smallest area, in the sense of Lebesgue measure, of a region $R \subset \R^2$ with the following property. The needle can be moved continuously inside $R$ so that it ends in its starting position with its two ends exchanged.

Remarks

The problem is solved, and the answer is that there is no positive minimum. Abram Besicovitch showed that for every $\varepsilon > 0$ there is a region of area less than $\varepsilon$ in which the needle can be turned around. His construction builds on a set of measure zero that contains a unit segment in every direction, an example of a Kakeya set. If $R$ is required to be convex, the minimum exists: it is the equilateral triangle of height $1$ , of area $1/\sqrt{3}$ , a result of Julius Pál.

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Tag	`WZ42B0SAVM`
URI	`m2://WZ42B0SAVM`
Kind	problem
Writing	W0 stub
MSC	28A75, 52A38
Status	This copy of the site has no build data. Run `m2 status WZ42B0SAVM`.
About	Kakeya set
Uses	Lebesgue measure

Kakeya needle problem

Problem

Remarks

Reading path

Further reading

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