In this talk, based on a joint work with Jean-Philippe Chassé, I will explain conditions under which a Lagrangian L admits a "Weinstein neighborhood of Hamiltonian (resp. symplectic) non-displacement", i.e. a neighborhood W such that any Lagrangian in the Hamiltonian (resp. symplectic) orbit of L, included in W, must intersect L. This result follows from exactness of nearby Lagrangians: any Lagrangian in the orbit of L, included in W, is exact in W seen as a subset of the cotangent bundle of L. Moreover, I will discuss several applications, in particular to the Lagrangian C⁰ flux conjecture and to the topology of orbits of Lagrangians.