Cube slicing in \({\mathbb{R}}^ n\). (English) Zbl 0601.52005
The main results of the paper can be formulated as follows: 1. Every central section of the unit cube \(Q\subset {\mathbb{R}}^ n\) by a hyperplane H has volume at least 1. The volume is 1 if and only if H is parallel to a (n-1)-dimensional face of Q. 2. Every central section of the unit cube \(Q\subset {\mathbb{R}}^ n\) by a hyperplane H has volume at most \(\sqrt{2}\). This upper bound is attained if and only if H contains an (n-2)- dimensional face of Q.
Reviewer: V.Soltan
MSC:
52A20 | Convex sets in \(n\) dimensions (including convex hypersurfaces) |
52A40 | Inequalities and extremum problems involving convexity in convex geometry |
60E05 | Probability distributions: general theory |