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]