Convex linear metric spaces are normable (Q2004382)

The authors call a linear metric space \((X,d)\) over the field \(\mathbb R\) convex if \(d(\lambda x+(1-\lambda)y,0)\le\lambda d(x,0)+(1-\lambda)d(y,0)\) holds for all \(x,y\in X\) and each \(\lambda\), \(0\le\lambda\le1\). They prove that each (real) convex linear metric space is normable (with the norm given by \(\|x\|=d(x,0)\). The same is true in the case of complex scalars if the metric \(d\) is rotation invariant (i.e., \(d(\alpha x,0)=d(|\alpha|x,0)\) for all complex scalars \(\alpha\) and all \(x\in X\)), but not in general.