2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/74709The variation distance closure of an exponential family with a convex set of canonical parameters is described, assuming no regularity conditions. The tools are the concepts of convex core of a measure and extension of an exponential family, introduced previously by the authors, and a new concept of accessible faces of a convex set. Two other closures related to the information divergence are also characterized.Published at http://dx.doi.org/10.1214/009117904000000766 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)Probability60A10 (Primary) 60B10\sep62B10\sep52A20 (Secondary)text