Parallelotopes of non-zero width

In 1959, Venkov introduced a concept of polytope of non-zero width in the direction of a subspace and studied parallelotopes of non-zero width. In the present paper properties of a parallelotope of non-zero width in the direction of a straight line are investigated. In particular, it is proved that a parallelotope of non-zero width in the direction of a straight line is the Minkowski sum of a parallelotope of width zero and a segment of this line. Necessary and sufficient conditions ensuring that the sum of a parallelotope and a line segment is again a parallelotope are presented.