A remark on the extremal non-central sections of the unit cube

Abstract

In this paper, we investigate extremal volumes of non-central slices of the unit cube. The case of central hyperplane sections is known and was studied by Ball, Hadwiger and Hensley. The case of non-central sections, i.e. when we dictate that the hyperplane must be a certain distance t > 0 from the center of the cube, is open in general and the same is true about sections of the unit cube by slabs. In this paper we give a full solution for extremal one-dimensional sections and a partial solution for extremal hyperplane slices for the case t > √n-1/2. We also make a remark on minimal volume slices of the cube by slabs of width 2t, when t > √n-1/2. © Springer Science+Business Media New York 2013.