An extension of Alexandrovʼs theorem on second derivatives of convex functions

If f is a function of n variables that is locally L1 approximable by a sequence of smooth functions satisfying local L1 bounds on the determinants of the minors of the Hessian, then f admits a second order Taylor expansion almost everywhere. This extends a classical theorem of A.D. Alexandrov, covering the special case in which f is locally convex.