Fix a positive integer $d$ and a positive number $s\leq d$. I am interested in calculating/estimating the area of the intersection between the hypercube $[0,1]^d$ and the hyperplane $\{\vec x\in \mathbb R^d, \sum_{i=1}^d x_i = s\}$. I wonder if there is any existing result on the area of this resulting polytope? I am mainly interested in the asymptotic regime whe $s = c d$ for fixed $c\in(0,1)$ and $d\rightarrow \infty$. Any other result/reference will also be appreciated.
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I don't know about the asymptotic behavior, but the hyper-area is $\frac{1}{(d-1)!}\sum\limits_{k=0}^d (-1)^k \binom{d}{k} (s - k)^{d-1}+$ where $t{+} = \max(t,0)$. The basic idea is figure out the area of the intersection of the hyperplane with $[0,\infty)^d$ and then apply inclusion exclusion principle. Loot at my answer to a similar question for 3-d to see what I mean. – achille hui May 31 '22 at 17:54
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In probabilistic terms, you are looking for the probability density function of $X_1+X_2+\ldots+X_d$, with $X_j$ being independent and uniformly distributed over $[0,1]$. It follows that the answer is given by the density function of the Irwin-Hall distribution, and for (moderately) large values of $d$ the asymptotics are given by the central limit theorem. Large sections occur for $s\in\left[\frac{d}{2}-c\sqrt{d},\frac{d}{2}+c\sqrt{d}\right]$ and the measure of the central section ($s=d/2$) is related to the integral of $\left(\frac{\sin x}{x}\right)^n$ over $\mathbb{R}^+$.
Jack D'Aurizio
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