On the number of antipodal or strictly antipodal pairs of points in finite subsets ofR d,II

The paper is a continuation of [MM], namely containing several statements related to the concept of antipodal and strictly antipodal pairs of points in a subsetX ofR^d, which has cardinalityn. The pointsx_i, x_jX are called antipodal if each of them is contained in one of two different parallel supporting hyperplanes of the convex hull ofX. If such hyperplanes contain no further point ofX, thenx_i, x_j are even strictly antipodal. We shall prove some lower bounds on the number of strictly antipodal pairs for givend andn. Furthermore, this concept leads us to a statement on the quotient of the lengths of longest and shortest edges of speciald-simplices, and finally a generalization (concerning strictly antipodal segments) is proved.Research (partially) supported by Hungarian National Foundation for Scientific Research, grant no. 1817