Abstract:We study the volume of central hyperplane sections of the cube. Using Fourier analytic and variational methods, we retrieve a geometric condition characterizing critical sections which, by entirely different methods, was recently proven by Ivanov and Tsiutsiurupa [Anal. Geom. Metr. Spaces 9 (2021), pp. 1-18]. Using this characterization result, we prove that critical central hyperplane sections in the 3-dimensional case are all diagonal to a (possibly lower dimensional) face of the cube, while in the 4-dimensional case, they are either diagonal to a face, or, up to permuting the coordinates and sign changes, perpendicular to the vector $(1,1,2,2)$. This shows the existence of non-diagonal critical central sections.

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中文翻译：

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立方体的关键中心部分

摘要：我们研究了立方体中心超平面截面的体积。使用傅里叶分析和变分方法，我们检索了表征关键部分的几何条件，通过完全不同的方法，最近由 Ivanov 和 Tsiutsiurupa [Anal. 几何。公尺。Spaces 9 (2021)，第 1-18 页]。使用这个表征结果，我们证明了 3 维情况下的关键中心超平面部分都与立方体的（可能较低维的）面成对角线，而在 4 维情况下，它们要么与面成对角线，要么，直到垂直于向量 $(1,1,2,2)$ 排列坐标和符号变化。这表明存在非对角线关键中心部分。

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