Regular hexagons in normed spaces and a theorem of Walter Benz (Q1801333)

Let \(E\) be a real normed linear space of dimension at least 3 and \(G\) be a map of \(E\) into \(E\) and \(p_ 0\) be a positive number such that \(G(x+h)-G(x)\) and \(h\) are linearly dependent for all vectors \(h\) with \(\| h\|=p_ 0\) and all vectors \(x\in E\). It is proven that then there are \(x_ 0\in E\) and a real number \(q_ 0\) such that \(G(x)=q_ 0 x+x_ 0\) for all \(x\in E\).