, Volume 3, Issue 1, pp 29-38

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

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Abstract

Letx 1,...,x m be points in the solid unit sphere ofE n and letx belong to the convex hull ofx 1,...,x m. 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(z 0)| where \(p(z) = \prod\limits_{i = 1}^m {(z - z_i ),\left| {z_i } \right| \leqq 1,i = 1,...m,} \) andp′(z 0)=0.