We investigate the computational complexity of computing the Hausdorff distance. Specifically, we show that the decision problem of whether the Hausdorff distance of two semi-algebraic sets is bounded by a given threshold is complete for the complexity class $${ \forall \exists _{<}\mathbb {R}} $$
∀
∃
<
R
. This implies that the problem is -, -, $$\exists \mathbb {R} $$
∃
R
-, and $$\forall \mathbb {R} $$
∀
R
-hard.