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

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.