  De Berg et al. in [SICOMP 2020] gave an algorithmic framework for
subexponential algorithms on geometric graphs with tight (up to ETH) running
times. This framework is based on dynamic programming on graphs of weighted
treewidth resulting in algorithms that use super-polynomial space. We introduce
the notion of weighted treedepth and use it to refine the framework of de Berg
et al. for obtaining polynomial space (with tight running times) on geometric
graphs. As a result, we prove that for any fixed dimension $d \ge 2$ on
intersection graphs of similarly-sized fat objects many well-known graph
problems including Independent Set, $r$-Dominating Set for constant $r$, Cycle
Cover, Hamiltonian Cycle, Hamiltonian Path, Steiner Tree, Connected Vertex
Cover, Feedback Vertex Set, and (Connected) Odd Cycle Transversal are solvable
in time $2^{O(n^{1-1/d})}$ and within polynomial space.
