On the Penrose inequality for dust null shells in the Minkowski spacetime of arbitrary dimension

A particular, yet relevant, case of the Penrose inequality involves null shells propagating in the Minkowski spacetime. Despite previous claims in the literature, the validity of this inequality remains open. In this paper, we rewrite this inequality in terms of the geometry of the surface obtained by intersecting the past null cone of the original surface S with a constant time hyperplane and the 'time height' function of S over this hyperplane. We also specialize to the case when S lies in the past null cone of a point and show the validity of the corresponding inequality in any dimension (in four dimensions this inequality was proved by Tod (1985 Class. Quantum Grav. 2 L65-8). Exploiting properties of convex hypersurfaces in the Euclidean space, we write down the Penrose inequality in the Minkowski spacetime of an arbitrary dimension n + 2 as an inequality for two smooth functions on the sphere Sn. We finally obtain a sufficient condition for the validity of the Penrose inequality in the four-dimensional Minkowski spacetime and show that this condition is satisfied by a large class of surfaces. (paper)