Minimal subsets of Borel Kakeya sets — Problem 4.2

4. Liang Yu

A set $A\subseteq\mathbb{R}^{2}$ is called a Kakeya set if, for every $\alpha\in\mathbb{R}$, there exists $C_{\alpha}\in\mathbb{R}$ such that

for every $(x,y)\in\mathbb{R}^{2}$. Equivalently,

for every $\alpha\in\mathbb{R}$.

A Kakeya set $A$ is called minimal if every proper subset $B\subsetneq A$ is not a Kakeya set.

Yinghe Peng has shown that there is a full fine Kakeya set which contains no minimal Kakeya set.

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