Let $n\ge 2$. Define $$Q:=\{x=(x_1, \ldots, x_n)\in \mathbb{R}^n: \max\limits_{j=1, \ldots, n} |x_j|\le \frac{1}{2}\}$$ and $$B=\{x=(x_1, \ldots, x_n)\in \mathbb{R}^n: \sqrt{\sum_{j=1}^n x_j^2}\le 1\}$$ What is the volume of $Q\cap B$?
For $n=2, 3, 4$, we see that $Q\cap B=Q$, so
$${\rm vol}(Q\cap B)=1$$
Moreover, for every $n\in \mathbb{N}$, we know that
$$
\tilde{Q}:=\{x=(x_1, \ldots, x_n)\in \mathbb{R}^n: \max\limits_{j=1, \ldots, n} |x_j|\le \frac{1}{\sqrt{n}}\} \subseteq Q\cap B,
$$
so we have a lower bound
$$
{\rm vol}(Q\cap B) \ge 2^n n^{-n/2}.
$$
Note that for $n=2$ the above inequality becomes equality, but for $n=3, 4$ we see that
$$
{\rm vol}(Q\cap B) > 2^n n^{-n/2}
$$
How do we estimate ${\rm vol}(Q\cap B)$? If we cannot compute the exact value of ${\rm vol}(Q\cap B)$, is the ``best" lower bound of ${\rm vol}(Q\cap B)$ known?