A Remark on the Extremal Non-central Sections of the Unit Cube

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 in [Ha], [Ba1] [He1]. 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, with some partial results provided in [BK], [KK1], [KK2]. 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 .