We characterize the topology of the space of Lorentzian polynomials with a given support in terms of the local Dressian. We prove that this space can be compactified to a closed Euclidean ball whose dimension is the rank of the Tutte group. Finally, we show that a compactification proposed by Brändén is in general not homeomorphic to a manifold with boundary. This is part of a joint work with Matt Baker, June Huh and Oliver Lorscheid.