Periodica Mathematica Hungarica

, Volume 27, Issue 3, pp 185–198 | Cite as

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

  • E. MakaiJr.
  • H. Martini
Article

Abstract

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 pointsxi, xj∈X 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, thenxi, xj 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.

Mathematics subject classification numbers, 1991

Primary 52C10 Secondary 52B12 

Key words and phrases

Mininimum number of strictly antipodar pairs of points quotient of the lengths of the maximal and minimal edges of a simplex strictly antipodal sets of segments 

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Copyright information

© Akadémiai Kiadó, Budapest 1993

Authors and Affiliations

  • E. MakaiJr.
    • 1
  • H. Martini
    • 2
  1. 1.Mathematical InstituteHungarian Academy of SciencesBudapestHungary
  2. 2.Mathematisches Institut, PH DresdenDresden

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