Lower bound on volume of $n$-cube intersected with $n$-sphere

Question

Let $B_n^r(c)$ be the radius $r$ ball in $\mathbb{R}^n$ dimensions centered at $c$. I am interested in $$\text{Vol}([-0.5, 0.5]^n \cap B_n^r(c)).$$ Is there a good lower bound for this quantity? I was able to find an exact formula for the concentric case (when $c = 0$), but it's not clear to me how to generalize this for arbitrary $c$.

The problem actually seems to be easier if we consider intersections with arbitrary rectangles. In this case, we can compute it recursively. Let $B$ be the unit ball in $\mathbb{R}^n$. Let $x, y \in \mathbb{R}^n$ and assume $x_i \geq 0, y_i \geq 0$ (for simplicity, as otherwise the argument is still similar but with extra casework). Consider the rectangle $R = \prod_{i=1}^n [x_i, x_i + y_i]$. Let $L = [-x_n - y_n, x_n + y_n] \times \prod_{i=1}^{n-1} [x_i, x_i + y_i]$ and let $S = [-x_n, x_n] \times \prod_{i=1}^{n-1} [x_i, x_i + y_i]$ Then we have $$\text{Vol}(R \cap B) = \frac{1}{2}\text{Vol}(L \cap B) - \frac{1}{2}\text{Vol}(S \cap B).$$ Note that $L, S$ are both centered on $0$ in the $n$-th coordinate. Repeating this process for each of $L, S$ we can reduce the problem to the concentric case, which is known. Can this be simplified into a closed form?

As Will noted in his answer, you can use probabilistic techniques to get good bounds for typical cases, but my application requires good lower bounds on tails, which I am not sure how to get.

Answer 1

For a bound, I would interpret this quantity as the probability that a random point $x$ in $[-0.5,0.5]^n$ lies in $B^r_n(c)$. If we let $x_1,\dots,x_n$ be the coordinates of $x$ and $c_1,\dots,c_n$ be the coordinates of $c$ then this is the probability that $\sum_{i=1}^n (x_i -c_i)^2 \leq r^2$.

Now $\sum_{i=1}^n (x_i -c_i)^2$ is a sum of independent, not necessarily identically distributed, random variables. Still, it will be approximately a Gaussian, centered on its mean $\frac{n}{12}+ \sum_{i=1}^n c_i^2$ and with variance, if I calculated correctly, $ \frac{n}{180} + \sum_{i=1}^n \frac{c_i^2}{3} $.

Depending on whether $r$ is greater than or less than this mean, the lower bound you want is either an upper bound or lower bound for the probability that this sum of random variables lies in a tail of the probability distribution. There are a lot of bounds for this kind of thing in the probability theory literature, and depending on $r$, $c$, and how precise a bound you need, one of them should be helpful.