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.


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