Convex Hulls and the Swampland

Several swampland constraints can be formulated in terms of geometrical convex hull conditions in the space of charges or in the field space of quantum gravity theories. In this talk, I will explain how the Distance conjecture can be related to a convex hull condition for the scalar charges of towers of particles. This can provide some global information about the moduli space and its dualities, as well as constrain the asymptotic scalar potentials and the amount of scale separation in AdS. I will also explain how the condition for not having accelerating cosmologies asymptotically can be formulated in terms of a similar but stronger convex hull condition applied to membranes.