An extension of Alexandrov's theorem on second derivatives of convex functions

If $f$ is a function of $n$ variables that is locally $L^1$ approximable by a sequence of smooth functions satisfying local $L^1$ 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.