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 $X$ be a locally convex space and $A,B\subset X$ nonempty closed convex sets. 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]

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References

J. Dieudonné (1966). "Sur la séparation des ensembles convexes". Math. Ann.. 163.
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.

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[1] J. Dieudonné (1966). "Sur la séparation des ensembles convexes". Math. Ann.. 163.

[Zalinescu-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.

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