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I've seen this statement of Radstrom cancellation: if $A +C \subset B+C$ where $A,B$ are convex, $B$ is closed, and $C$ is bounded, then $A \subset B.$

Is it essential that $A$ be convex?

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For those who are on first-read as confused as I was, A,B,C are subsets of a given normed vector space, and I assume that + is the addition of sets. – Theo Johnson-Freyd Dec 15 '09 at 5:32

No. Actually, the original version of the lemma does not require the convexity of A.

Reference: Hans Rådström (note spelling): An embedding theorem for spaces of convex sets. Proc. Amer. Math. Soc. 3 (1952) 165–169, MR0045938. doi:10.2307/2032477.

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Alex, your idea of using a separation theorem can be turned into a full proof as follows:

As you said, $A \not\subset B$ iff there is a $a \in A$ and vector $x$ such that $\sup_{b \in B} \langle b,x \rangle < \langle a,x \rangle$. But then,

\begin{align} \sup_{z \in B + C} \langle z, x \rangle &= \sup_{b \in B, c \in C} (\langle b, x \rangle + \langle c, x \rangle ) \\\\ &= \sup_{b \in B} \langle b, x \rangle + \sup_{c \in C} \langle c, x \rangle \\\\ &< \langle a, x \rangle + \sup_{c \in C} \langle c, x \rangle \\\\ &= \sup_{z \in a + C} \langle z, x \rangle \\\\ &\leq \sup_{z \in B + C} \langle z, x \rangle, \end{align}

a contradiction.

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Hmm, I have an idea for a proof which works even when $A$ isn't convex:

$A \not\subset B$ iff there is a $a \in A$ and vector $x$ such that $\langle a, x\rangle >0$ and $\langle b, x \rangle \leq 0$ for all $b \in B$ (since $B$ is a closed convex set). Let $b \in B$ and $c \in C$ maximize $\langle b + c, x \rangle$, then $\langle a + c, x \rangle > \langle b + c, x \rangle$, so there is a point in $A + C$ that isn't in $B+C$.

This implies that if $A \not\subset B$ then $A + C \not\subset B+C$. Of course, there aren't really $b,c$ which maximize that quantity above, but I believe you could make this rigorous using approximation arguments.

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Replace $A$ by its convex hull and use the result for convex $A$.

Edit: as observed in the comments, this is not enough.

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Maybe I'm missing something but why does the assumption of the original statement then hold? – Reid Barton Dec 15 '09 at 4:02
Because the only assumption on A in the original statement is that it is convex, and its conclusion surely implies what one wants. – Mariano Suárez-Alvarez Dec 15 '09 at 4:15
It seems to me that you can replace both A and C by their respective convex hulls and get the desired result. – Harald Hanche-Olsen Dec 15 '09 at 4:17
@Mariano: The problem is that B+C may not be convex. – Harald Hanche-Olsen Dec 15 '09 at 4:18
I am then the one missing something... that B+C be convex is not an hypothesis of the original statement (and in any case is a condition independent of $A$!) – Mariano Suárez-Alvarez Dec 15 '09 at 4:21

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