Let $B_\infty = [-1, 1]^n$, and let $B(x, r) = \{y \in \mathbb{R}^n : \|x - y\|\ \leq r\}$. I would like to bound (or compute, if possible)
$$ v(x, r) := \lambda(B_\infty \cap B(x, r)) $$ as a function of $x \in B_\infty$ and $r > 0$. Above $\lambda$ denotes Lebesgue measure in $\mathbb{R}^n$.
Of course, if $r$ is large enough (say $r \geq 1 + \sqrt{n}$), then $B_\infty \cap B(x, r) = B_\infty$, and thus $v(x, r) = 2^n$. So, it is only for somewhat small $r$ say $r \ll \sqrt{n}$, that I would like to get such a bound. The naive estimate would be $$ v(x, r) \leq \min\left\{2^n, \frac{r^n\pi^{n/2}}{\Gamma(n/2 + 1)}\right\}, $$ simply by using the two volumes together. Is this the best possible estimate?