正方体截面

guanmo1

求证：给定正方体的任意截面中，体对角面的面积最大。

hbghlyj

mathoverflow.net/questions/247882  the maximal area is $\sqrt{2}$ it is attained only if the hyperplane has a normal unit vector with two nonzero coordinates with absolute value $1/\sqrt2$ As far as I know this is Ball's theorem. Fedja wrote a simple proof once many years ago math.spbu.ru/analysis/f-doska/cube.pdf Pooley2012.pdf (635.45 KB, Downloads: 40) 2023-4-14 20:30 Upload Click on the file to download the attachment

guanmo1

hbghlyj 发表于 2023-4-14 20:10

谢谢您。求初等证明。

hbghlyj

mathoverflow.net/questions/430553/
Let $1<n\in\mathbb N,$ $S(n)$ is the greatest $(n-1)$-area of $L\cap I^n$ where $I^n\subseteq\mathbb R^n$ is the unit cube, and $L$ runs over all possible affine $(n-1)$-hyperplanes.

For all $n>1$, $S(n)=\sqrt2$, it is a result of Keith Ball
PDF K. M. Ball. Cube slicing in Rⁿ. Proc Amer Math Soc 97:3 (1986), pp. 465-473

guanmo1

hbghlyj 发表于 2023-4-15 05:43
https://www.zhihu.com/question/389872846
https://www.zhihu.com/question/490004301/answer/2798990583
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