To be short, I have an hypercube $C_\delta = [0, \delta]^{n+1}$ and an hyperplane $H_t = \{x \in \mathbb{R}^{n+1} : \|x\|_1 = t\}$. The quantity I'm looking for is the hyper-area of $C_\delta \cap H_1$, and I assume $1/(n+1) \leq \delta \leq 1$ to ensure the intersection is non-null. I have seen other topics with very similar questions, however they use the unit hypercube $C_1$.
If you want some context, I was trying to determine a probability distribution when I came to the conclusion that it was equal to the aforementioned quantity.
I know the quantity to be equal -or at least very similar- to $\lambda_n(C_\delta \cap H_1) = \sum_{k=0}^{n+1} (-1)^k \binom{n+1}{k}(1-\delta k)^n 1_{\delta k \leq 1}$, however I struggle to come up with an proper proof.
I have tried the following :
Using a geometrical approach I found in James E. Marengo, David L. Farnsworth, Lucas Stefanic, "A Geometric Derivation of the Irwin-Hall Distribution", International Journal of Mathematics and Mathematical Sciences, vol. 2017, Article ID 3571419, 6 pages, 2017. https://doi.org/10.1155/2017/3571419 in the proof of Theorem 1 of Section 2. However it is not exactly the case I want and I can't quite see how to cross the gap to get to mine. My problem being the evaluation of the volume of the intersection of the $B_j(t)$ when I want the coordinates to be $>\delta$ instead of $>1$, else it would be an already answered question. It would give the hypervolume of the intersection between $C_\delta$ and the half-space $\{x \in \mathbb{R}^{n+1} : \|x\|_1 \leq t\}$. I then need only compute its derivative with respect to $t$ and evaluate it for $t=1$.
Using a naive approach with the Irwin-Hall distribution I detailed in my answer to my initial problem. It gave me a function that didn't match my formula in a very specific case, but tweaking it a bit -namely : multiplying by $\delta^{n+1} n!$- made it match and works marvellously with empiric testing. While I see where the $\delta^{n+1}$ could come from, I have no clue for the $n!$.
Do you have any idea on how to proceed further ?
