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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?2012-08-14
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    There is an equality symbol instead of an inclusion, which makes this statement stronger2012-08-14
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    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.2012-08-14
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    I've now fixed [my answer to the other question](http://math.stackexchange.com/a/175016/6622).2012-08-18

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