Alexandrov theorem

In mathematical analysis, the Alexandrov theorem, named after Aleksandr Danilovich Aleksandrov, states that if U is an open subset of [math]\displaystyle{ \R^n }[/math] and [math]\displaystyle{ f\colon U\to \R^m }[/math] is a convex function, then [math]\displaystyle{ f }[/math] 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.

References

• Niculescu, Constantin P.; Persson, Lars-Erik (2005). Convex Functions and their Applications: A Contemporary Approach. Springer-Verlag. p. 172. ISBN 0-387-24300-3. 
• Villani, Cédric (2008). Optimal Transport: Old and New. Grundlehren Der Mathematischen Wissenschaften. 338. Springer-Verlag. p. 402. ISBN 978-3-540-71049-3.