A spacelike surface in a Lorentzian four-manifold is said to be marginally trapped if it mean curvature vector is null (lightlike). Marginally trapped surfaces play an important role in general relativity: they describe the horizon of black holes. However, their mathematical properties are still poorly understood, although marginally trapped surfaces satisfying several additional properties have recently been characterized: in 2006, B.-Y. Chen and F. Dillen classied marginally trapped Lagrangian surfaces in complex Lorentzian space forms; in 2007 J. Van der Veken described those marginally trapped surfaces of Lorentzian space-forms which have positive relative nullity; in 2009 the classication of marginally trapped surfaces of constant curvature in complex Lorentzian space forms was performed by B.-Y. Chen. In a joint work with Yamile Godoy (Cordoba University, Argentina), we take advantage of the natural contact structure enjoyed by the space of null geodesics of a $(n + 1)$-dimensional Lorentzian manifold, recently uncovered by B. Khesin/S. Tabachnikov and Y. Godoy/M. Salvai: it turns out that the congruence of the null geodesics normal to a $(n - 1)$-dimensional spacelike submanifold is a Legendrian submanifold. This allows to give an explicit, local description of $(n-1)$-dimensional, marginally trapped submanifolds in Lorentzian space forms, and in $\mathbb{S}^{n+1}\times\mathbb{R}$ and $\mathbb{H}^{n+1}\times\mathbb{R}$: We shall also discuss recent progress about marginally trapped surfaces in Euclidean four-space endowed with the neutral, flat metric.

The width of a closed convex subset of the Euclidean space is the distance between two parallel supporting planes. For example the sphere has constant width in all directions, but there is an infinity of convex subsets sharing this property. The Blaschke-Lebesgue problem, consisting of minimizing the volume in the class of convex sets of fixed constant width is still open. In this talk we shall describe a necessary condition that must satisfy the minimizer of the Blaschke-Lebesgue.