  The paper establishes a sharp and rigid isoperimetric-type inequality in
Lorentzian signature under the assumption of Ricci curvature bounded below in
the timelike directions. The inequality is proved in the high generality of
Lorentzian pre-length spaces satisfying timelike Ricci lower bounds in a
synthetic sense via optimal transport, the so-called $\mathsf{TCD}^e_p(K,N)$
spaces. The results are new already for smooth Lorentzian manifolds.
Applications include an upper bound on the area of Cauchy hypersurfaces inside
the interior of a black hole (original already in Schwarzschild) and an upper
bound on the area of Cauchy hypersurfaces in cosmological space-times.
