Distance between finite sets of points

Question

Denote the collection of all finite subsets in $\mathbb{R}^d$ as $\mathcal{S} = \{S \subseteq \mathbb{R}^d: |S| < \infty \}$. What are ways to define distance metrics on $\mathcal{S}$ that can be efficiently computed? For instance, one could define $$ d(A, B) = \frac{1}{|A|\cdot|B|} \sum_{x\in A}\sum_{y\in B} \|x - y\|_2^2 $$ but I'm not sure if the triangle inequality is satisfied?