论文标题
$ n $ -cube的中央对角线部分
Central diagonal sections of the $n$-cube
论文作者
论文摘要
我们证明，在$ \ mathbb {r}^n $正交中，单位立方体的中央超平面部分与立方体的直径是$ n \ geq 3 $的尺寸的严格单调增加的功能。我们的参数使用一个积分公式，该公式可以返回到pólya\ cite {p}（另请参见\ cite {h}和\ cite {b86}），以获取立方体的中央部分的体积，以及Laplace的方法来估算整体的渐进性行为。首先，我们证明单调性从一些特定的$ n_0 $开始。然后，使用间隔算术（IA）和自动分化（AD），我们计算了以$ N_0 $的明确绑定，并通过直接计算检查其余案例之间的剩余案例。
We prove that the volume of central hyperplane sections of a unit cube in $\mathbb{R}^n$ orthogonal to a diameter of the cube is a strictly monotonically increasing function of the dimension for $n\geq 3$. Our argument uses an integral formula that goes back to Pólya \cite{P} (see also \cite{H} and \cite{B86}) for the volume of central sections of the cube, and Laplace's method to estimate the asymptotic behaviour of the integral. First we show that monotonicity holds starting from some specific $n_0$. Then, using interval arithmetic (IA) and automatic differentiation (AD), we compute an explicit bound for $n_0$, and check the remaining cases between $3$ and $n_0$ by direct computation.
