Consider non-empty sets $\mathcal{A},\mathcal{B},\mathcal{C} \subseteq \mathbb{R}^n$. The Hausdorff distance between $\mathcal{A}$ and $\mathcal{B}$ is
$$d_H(\mathcal{A},\mathcal{B}) := \max\left\{\sup_{a \in \mathcal{A}} \inf_{b \in \mathcal{B}} \|a - b\|,~\sup_{b \in \mathcal{B}} \inf_{a \in \mathcal{A}} \|a - b\|\right\}.$$
The Minkowski sums $\mathcal{A} + \mathcal{C}$ and $\mathcal{B} + \mathcal{C}$ are
$$ \begin{split} \mathcal{A} + \mathcal{C} &= \left\{a + c\colon a \in \mathcal{A}, c\in \mathcal{C}\right\},\\ \mathcal{B} + \mathcal{C} &= \left\{b + c\colon b \in \mathcal{B}, c\in \mathcal{C}\right\}. \end{split} $$
Then, does $d_H(\mathcal{A}+\mathcal{C},\mathcal{B}+\mathcal{C}) = d_H(\mathcal{A},\mathcal{B})$ hold? Thanks very much!