Quantitative Lorentzian isoperimetric inequalities

We establish optimal stability estimates in terms of the Fraenkel asymmetry with universal dimensional constants for a Lorentzian isoperimetric inequality due to Bahn and Ehrlich and, as a consequence, for a special version of a Lorentzian isoperimetric inequality due to Cavalletti and Mondino. For the Bahn--Ehrlich inequality the Fraenkel asymmetry enters the stability result quadratically like in the Euclidean case while for the Cavalletti--Mondino inequality the Fraenkel asymmetry enters linearly. As it turns out, refining the latter inequality through an additional geometric term allows us to recover the more common quadratic stability behavior. Along the way, we provide simple, self-contained proofs for the above isoperimetric-type inequalities. Moreover, in a fixed conical Minkowski spacetime, we use a Lipschitz bound, naturally provided by the causal structure, to upgrade our quantitative control to a Hausdorff stability estimate. This estimate is formulated in terms of a distance defined by Bahn and Ehrlich, which restricts to a natural Hausdorff-type metric on the space of Cauchy hypersurfaces.