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 $A,B \subset X$ a locally convex space, if either $A$ or $B$ is locally compact and $\operatorname{recc}(A) \cap \operatorname{recc}(B)$ (where $\operatorname{recc}$ gives the recession cone) is a linear subspace, then $A - B$ 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.