# Dieudonné's theorem

In mathematics, Dieudonné's theorem, named after Jean Dieudonné, is a theorem on when the Minkowski sum of closed sets is closed.

## Statement of theorem

Let nonempty closed convex sets ${\displaystyle A,B\subset X}$ a locally convex space, if either ${\displaystyle A}$ or ${\displaystyle B}$ is locally compact and ${\displaystyle \operatorname {recc} (A)\cap \operatorname {recc} (B)}$ (where ${\displaystyle \operatorname {recc} }$ gives the recession cone) is a linear subspace, then ${\displaystyle A-B}$ is closed.[1][2]

## References

1. ^ J. Dieudonné (1966). "Sur la séparation des ensembles convexes". Math. Ann.. 163.
2. ^ Zălinescu, Constantin (2002). Convex analysis in general vector spaces. River Edge, NJ: World Scientific Publishing Co., Inc. pp. 6–7. ISBN 981-238-067-1. MR 1921556.