Israel Journal of Mathematics

, Volume 3, Issue 1, pp 29–38

On the product of the distances of a point from the vertices of a polytope

  • Binyamin Schwarz

DOI: 10.1007/BF02760025

Cite this article as:
Schwarz, B. Israel J. Math. (1965) 3: 29. doi:10.1007/BF02760025


Letx1,...,xm be points in the solid unit sphere ofEn and letx belong to the convex hull ofx1,...,xm. Then\(\prod\limits_{i = 1}^m {\left| {x - x_i } \right.\left\| \leqq \right.(1 - \left\| x \right\|)(1 + \left\| x \right\|)m^{ - 1} } \). This implies that all such products are bounded by (2/m)m(m −1)m−1. Bounds are also given for other normed linear spaces. As an application a bound is obtained for |p(z0)| where\(p(z) = \prod\limits_{i = 1}^m {(z - z_i ),\left| {z_i } \right| \leqq 1,i = 1,...m,} \) andp′(z0)=0.

Copyright information

© Hebrew University 1965

Authors and Affiliations

  • Binyamin Schwarz
    • 1
  1. 1.Technion - Israel Institute of TechnologyHaifa

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