Quantum Position Verification (QPV) schemes use the properties of quantum mechanics and the relativistic signalling bound to verify the location of a prover to distant verifiers. Recently, QPV has attracted considerable attention, with proof of principle experimental demonstrations and connections with several unexpected topics, such as holographic quantum gravity or quantum Hamiltonian simulation. Despite that, it is still an open question whether unconditionally secure QPV protocols exist, in the sense that the resources required to break them by cooperating adversaries scale exponentially. In this talk, I will show how Banach space geometry gives us a roadmap to address that question.