Diameters of convex bodies

In [2], we have shown that the extended diameter family of a convex body C in n-space has the property of outward simplicity if and only if there is no pair of parallel line segments in the boundary of C respectively in an opposite pair of parallel contact planes of C. In [3] Professor Sobczyk and I show that every outwardly simple family of lines in the plane is the extended diameter family of a constant breadth convex body and incidentally give a simple construction for all constant breadth planar convex bodies. The purpose of this note is to show that not every continuous outwardly simple line family in Euclidean 3-space is the extended diameter family of a convex body. A diameter of a convex body in En (n > 2) is a longest chord in its direction. In [2] it is proved that the diameters of a convex body cover the body.