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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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    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 $