I have a closed hypercube $Q_j$ for all $j\ge 1$ and I would like to find a open hypercube $S_j$ such that $$\vert S_j\vert\le (1+\varepsilon)\vert Q_j\vert. \quad(1)$$
I am not sure how can I do this, for $n=2$ we have a square $Q=[a,b]\times[c,d]$ and when I draw a picture I imagine doing the following procedure to get $S=$
$$(a,c)\to(a+\varepsilon,c+\varepsilon)$$ $$(b,c)\to(b-\varepsilon,c+\varepsilon)$$ $$(a,d)\to(a+\varepsilon,d-\varepsilon)$$ $$(b,d)\to(b-\varepsilon,d-\varepsilon)$$
But I don't get $(1).$
I imagine perhaps a more "abstract" argument ?
You can do it in each dimension separately. In one dimension you start with $(a,b)$ and want the length to become $(1+\epsilon)(b-a)$, so the segment becomes $(a-\frac \epsilon 2(b-a),b+\frac \epsilon 2(b-a))$. This expands it equally around the center.