# Kakeya Problem

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## 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)