Alexandrov theorem

In

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.
  Zbl 1100.26002
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• Zbl 1156.53003
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