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

TL;DR

This paper extends Alexandrov's theorem by showing that functions approximable by smooth functions with bounded Hessian minors almost everywhere admit a second order Taylor expansion, broadening the class of functions with second derivatives.

Contribution

It generalizes Alexandrov's theorem to functions that are not necessarily convex but can be approximated by smooth functions with controlled Hessian minors.

Findings

Functions approximable by smooth functions with bounded Hessian minors have second derivatives almost everywhere.

The result extends classical convex analysis to a broader class of functions.

Almost everywhere second order Taylor expansion is guaranteed under the given approximation conditions.

Abstract

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.