  We show that for any sufficiently rich compact family $\mathcal{H}$ of $C^1$
diffeomorphisms of a closed Riemannanian manifold $M$, the average geometric
intersection number over $h \in \mathcal{H}$ between $h(V)$ and $W$, for $V, W$
any complementary dimensional submanifolds of $M$, is approximately (i.e. up to
a uniform multiplicative error depending only on $\mathcal{H}$) the product of
their volumes. We also give a construction showing that such families always
exist.
