Abstracts > Gonzalez Cazares Jorge

We investigate the geometric regularity of the closed convex hull of a multidimensional Lévy process in $\R^d$ ($d \ge 2$). Resolving an open question regarding boundary smoothness beyond Brownian motion, we identify a broad class of Lévy processes, including $\alpha$-stable processes with full support and processes of arbitrarily low activity, whose convex hull boundary almost surely has no corners in any dimension, implying complete smoothness in $d=2$. These regularity properties extend to paths under locally diffeomorphic transformations and preclude two-sided corner or thorn points. Conversely, we prove that finite variation processes with a non-zero natural drift almost surely exhibit corners at directional extrema. To establish these results, we conduct a detailed local analysis of normal cones and exposed faces. Our proofs rely on multidimensional extensions of celebrated fluctuation identities, a stick-breaking representation of the convex keel, and a novel subsampling approach to analyse local behaviour at directional extrema via time-changing the process using inverse local times at directional records.