\begin{definition}[Definition:Kakeya's Constant]
'''Kakeya's constant''' is defined as the area of the smallest simple convex domain in which one can put a line segment of length $1$ which will coincide with itself when rotated $180 \degrees$:
:$K = \dfrac {\paren {5 - 2 \sqrt 2} \pi} {24} \approx 0 \cdotp 28425 \, 82246 \ldots$
{{OEIS|A093823}}
{{expand|Needs considerable work done here by someone who understands exactly what is going on here. The case of the equilateral triangle is well known; so is the case of the Perron tree; I also remember a piece by Martin Gardner on the subject which demonstrates that a star-shaped area derived from the deltoid can be made arbitrarily small; and so on. Exactly what is meant here by simple convex domain needs rigorous clarification.}}
\end{definition}