Alexandrov theorem explained

In mathematical analysis, the Alexandrov theorem, named after Aleksandr Danilovich Aleksandrov, states that if is an open subset of

\R ^{n} and

f\colon U\to \R ^{m} is a convex function, then 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

• Book: Convex Functions and their Applications: A Contemporary Approach . Niculescu . Lars-Erik . Persson . . 2005 . 0-387-24300-3 . 1100.26002 . 172 .
• Book: Villani, Cédric . Cédric Villani

. Optimal Transport: Old and New . 338 . Grundlehren Der Mathematischen Wissenschaften . Cédric Villani . . 2008 . 978-3-540-71049-3 . 1156.53003 . 402 .