A Kakeya set in the linear representation $T^{ *}_{ 2}(\mathcal{ C})$, $\mathcal{ C}$ a non-singular conic, is the point set covered by a set of $q+1$ lines, one through each point of $\mathcal{ C}$. In this article we classify the small Kakeya sets in $T^{ *}_{ 2}(\mathcal{ C})$. The smallest Kakeya sets have size $\left\lfloor\frac{ 3q^{ 2}+2q}{ 4}\right\rfloor$, and all Kakeya sets with weight less than $\left\lfloor\frac{ 3(q^{ 2}-1)}{ 4}\right\rfloor+q$ are classified: there are approximately $\sqrt{ \frac{ q}{ 2}}$ types.
Categories: math.CO