  We investigate optimal geographical caching in heterogeneous cellular
networks, where different types of base stations (BSs) have different cache
capacities. The content library contains files with different popularities. The
performance metric is the total hit probability.
  The problem of optimally placing content in all BSs jointly is not convex in
general. However, we show that when BSs are deployed according to homogeneous
Poisson point processes (PPP), independently for each type, we can formulate
the problem as a convex problem. We give the optimal solution to the joint
problem for PPP deployment. For the general case, we provide a distributed
local optimization algorithm (LOA) that finds the optimal placement policies
for different types of BSs. We find the optimal placement policy of the small
BSs (SBSs) depending on the placement policy of the macro BSs (MBSs). We show
that storing the most popular content in the MBSs is almost optimal if the SBSs
are using an optimal placement policy. Also, for the SBSs no such heuristic can
be used; the optimal placement is significantly better than storing the most
popular content. Finally, we numerically verify that LOA gives the same hit
probability as the joint optimal solution for the PPP model.
