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