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 $$A+C_0=B+C_0\implies A=B?$$