Kakeya Problem

Problem

The Kakeya Problem is the question:

What is the smallest possible area of a set in the plane inside which a needle of length $1$ can be moved continuously in order to reverse its direction?

Solution

There is no such smallest area.

That is, let $\epsilon \in \R_{>0}$ be a (strictly) positive real number.

Then there exists a plane figure which fulfils the conditions of the Kakeya Problem whose area is less than $\epsilon$.

Proof

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Also known as

The Kakeya Problem can also be seen rendered as Kakeya's Problem.

Source of Name

This entry was named for Soichi Kakeya.

Historical Note

The Kakeya Problem was raised by Soichi Kakeya in $1917$.

It remained unanswered until $1928$, at which time Abram Samoilovitch Besicovitch solved it.

Sources

• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): Kakeya's problem (S. Kakeya, 1917)