Editing Convex set (section)

=== Minkowski sums of convex sets ===
The Minkowski sum of two compact convex sets is compact. The sum of a compact convex set and a closed convex set is closed.<ref>Lemma&nbsp;5.3: {{cite book|first1=C.D.|last1= Aliprantis|first2=K.C.| last2=Border|title=Infinite Dimensional Analysis, A Hitchhiker's Guide| publisher=Springer| location=Berlin|year=2006|isbn=978-3-540-29587-7}}</ref>

The following famous theorem, proved by Dieudonné in 1966, gives a sufficient condition for the difference of two closed convex subsets to be closed.<ref name="Zalinescu p. 7">{{cite book|last=Zălinescu|first=C.|title=Convex analysis in general vector spaces|url=https://archive.org/details/convexanalysisge00zali_934|url-access=limited|publisher=World Scientific Publishing Co., Inc|location= River Edge, NJ |date= 2002|page=[https://archive.org/details/convexanalysisge00zali_934/page/n27 7]|isbn=981-238-067-1|mr=1921556}}</ref> It uses the concept of a '''recession cone''' of a non-empty convex subset ''S'', defined as:
<math display=block>\operatorname{rec} S = \left\{ x \in X \, : \, x + S \subseteq S \right\},</math>
where this set is a [[convex cone]] containing <math>0 \in X </math> and satisfying <math>S + \operatorname{rec} S = S</math>. Note that if ''S'' is closed and convex then <math>\operatorname{rec} S</math> is closed and for all <math>s_0 \in S</math>,
<math display=block>\operatorname{rec} S = \bigcap_{t > 0} t (S - s_0).</math>

'''Theorem''' (Dieudonné). Let ''A'' and ''B'' be non-empty, closed, and convex subsets of a [[locally convex topological vector space]] such that <math>\operatorname{rec} A \cap \operatorname{rec} B</math> is a linear subspace. If ''A'' or ''B'' is [[locally compact]] then ''A''&nbsp;−&nbsp;''B'' is closed.