Let $\mathbb{R}^N$ be a D816: Euclidean real Cartesian product.
Let $\mathsf{Vol}$ be the D1738: Euclidean real volume function on $\mathbb{R}^N$.
Let $I \subseteq \mathbb{R}^N$ be D3036: Real N-interval.

Then \begin{equation} \forall \, x \in \mathbb{R}^N : \mathsf{Vol} (I + x) = \mathsf{Vol} (I) \end{equation}