Regular hexagons in normed spaces and a theorem of Walter Benz

Abstract

A new proof is given for the following recent result of W. Benz [1]. Suppose thatG: E → E, whereE is a real normed vector space of dimension at least 3, is such that, for somep o > 0, $$G(x + h) - G(x)\,and h\,\,are linearly dependent whenever\,\,\left\| h \right\| = p_0 .$$ Then there existx o ∈ E andq o ∈ ℝ such thatG(x) = q 0x + x0 for allx ∈ E. The new proof is based on existence theorems for regular hexagons in normed spaces which may be of some interest in themselves. Hexagons are also helpful in deducing similar results for certain two-dimensional spaces.