In the seminar, I will present recent joint work with Fabio Cavalletti (Milan), establishing a sharp and rigid isoperimetric-type inequality in Lorentzian signature under the assumption of Ricci curvature bounded below in the time-like directions. The inequality is proved in the high generality of Lorentzian length spaces satisfying time-like Ricci curvature lower bounds in a synthetic sense via optimal transport. The results are new already for smooth Lorentzian manifolds. Applications include an upper area bound of Cauchy hypersurfaces inside the interior of a black hole and an upper bound on the area of Cauchy hypersurfaces in cosmological space-times.