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

Let ${\displaystyle X}$ be a locally convex space and ${\displaystyle A,B\subset X}$ nonempty closed convex sets. 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: 1–3. doi:10.1007/BF02052480. S2CID 119742919.
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.