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I recently found a question about a property of the Minkowski sums. However the question was not properly answered (it used a projection argument which might not be true in a general Banach space).

I was wondering whether the following (weaker) statement holds:

Let $X$ be a Banach space and suppose $A,B,C_0\subset X$ are bounded, closed, convex and non-empty subset. Do we then have $$A+C_0=B+C_0\implies A=B?$$

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what is the difference with that question? – Norbert Aug 14 '12 at 20:03
There is an equality symbol instead of an inclusion, which makes this statement stronger – gifty Aug 14 '12 at 20:12
This statement is weaker, not stronger. Yours immediately follows from the other one (as shown in the answer) while the other one doesn't follow from yours. I find it objectionable to call the other question not properly answered, as the easy fix was presented in a comment while joriki's answer gave the important geometric intuition. – t.b. Aug 14 '12 at 20:34
I've now fixed my answer to the other question. – joriki Aug 18 '12 at 4:55
up vote 2 down vote accepted

Modulo result presented in this question the solution is extremely simple $$ A+C_0=B+C_0\Longleftrightarrow (A+C_0\subset B+C_0)\wedge(B+C_0\subset A+C_0)\Longrightarrow $$ $$ (A\subset B)\wedge (B\subset A)\Longleftrightarrow A=B $$

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