Alexandrov theorem

In mathematical analysis, the Alexandrov theorem, named after Aleksandr Danilovich Aleksandrov, states that if {{mvar|U}} 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

{{cite book | title=Convex Functions and their Applications: A Contemporary Approach | first=Constantin P. | last1=Niculescu | first2=Lars-Erik | last2=Persson | publisher=Springer-Verlag | year=2005 | isbn=0-387-24300-3 | zbl=1100.26002 | page=172 }}
{{cite book | title=Optimal Transport: Old and New | volume=338 | series=Grundlehren Der Mathematischen Wissenschaften | authorlink=Cédric Villani | first=Cédric | last=Villani | publisher=Springer-Verlag | year=2008 | isbn=978-3-540-71049-3 | zbl=1156.53003 | page=402 }}

Category:Theorems in measure theory