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Let $X$ be a Banach space and endow the space $BC(X)$, the space of all bounded closed subsets of $X$, with the Hausdorff distance $d_H$. Fix $C_0\in BC(X)$. Is it true that $d_H(A,B)=d_H(A+C_0,B+C_0)$ for all $A,B\in BC(X)$?

Edit: Is it true when $A,B,C_0$ are convex sets?

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