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.