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On the Volume of Sections of the Cube
Analysis and Geometry in Metric Spaces ( IF 0.9 ) Pub Date : 2021-01-01 , DOI: 10.1515/agms-2020-0103
Grigory Ivanov 1 , Igor Tsiutsiurupa 2
Affiliation  

We study the properties of the maximal volume k -dimensional sections of the n -dimensional cube [−1, 1] n . We obtain a first order necessary condition for a k -dimensional subspace to be a local maximizer of the volume of such sections, which we formulate in a geometric way. We estimate the length of the projection of a vector of the standard basis of ℝ n onto a k -dimensional subspace that maximizes the volume of the intersection. We find the optimal upper bound on the volume of a planar section of the cube [−1, 1] n , n ≥ 2.

中文翻译:

关于立方体截面的体积

我们研究n维立方体[−1,1] n的最大体积k维截面的性质。我们为ak维子空间成为此类截面的体积的局部极大值提供了一阶必要条件,我们以几何方式对其进行了公式化。我们估计以n为标准单位的向量到ak维子空间上的投影长度,该向量会最大化交集的体积。我们在立方体[-1,1] n,n≥2的平面部分的体积上找到最佳上限。
更新日期:2021-01-01
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