  Kleiner-Leeb and Quint showed that convex subsets in higher-rank symmetric
spaces are very rigid compared to rank 1 symmetric spaces. Motivated by this,
we consider convex subsets in products of proper CAT(0) spaces $X_1\times X_2$
and show that for any two convex co-compact actions $\rho_i(\Gamma)$ on $X_i$,
where $i=1, 2$, if the diagonal action of $\Gamma$ on $X_1\times X_2$ via
$\rho=(\rho_1, \rho_2)$ is also convex co-compact, then under a suitable
condition, $\rho_1(\Gamma)$ and $\rho_2(\Gamma)$ have the same marked length
spectrum.
