mathematical analysis Analysis is the branch of mathematics dealing with continuous functions, limits, and related theories, such as differentiation, integration, measure, infinite sequences, series, and analytic functions. These theories are usually studied ...

, the Alexandrov theorem, named after

Aleksandr Danilovich Aleksandrov Aleksandr Danilovich Aleksandrov (russian: Алекса́ндр Дани́лович Алекса́ндров, alternative transliterations: ''Alexandr'' or ''Alexander'' (first name), and ''Alexandrov'' (last name)) (4 August 1912 – 27 July 19 ...

, 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 In mathematical analysis, Rademacher's theorem, named after Hans Rademacher, states the following: If is an open subset of and is Lipschitz continuous, then is differentiable almost everywhere in ; that is, the points in at which is ''not'' di ...

References

* * Theorems in measure theory {{mathanalysis-stub