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.

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. Zbl 1100.26002.
• Villani, Cédric (2008). Optimal Transport: Old and New. Grundlehren Der Mathematischen Wissenschaften. 338. Springer-Verlag. p. 402. ISBN 3-540-71049-3. Zbl 1156.53003.