Alexandrov theorem

In mathematical analysis, the Alexandrov theorem, named after Aleksandr Danilovich Aleksandrov, states that if U is an open subset of ${\displaystyle \mathbb {R} ^{n}}$ and ${\displaystyle f\colon U\to \mathbb {R} ^{m}}$ is a convex function, then ${\displaystyle f}$ has a second derivative almost everywhere.

In this context, having a second derivative at a point means having a second-order Taylor expansion at that point with a local error smaller than any quadratic.

The result is closely related to Rademacher's theorem.