  We study geometric properties of GL-spaces. We demonstrate that every
finite-dimensional GL-space is polyhedral; that in dimension 2 there are only
two, up to isometry, GL-spaces, namely the space whose unit sphere is a square
(like $\ell_\infty^2$ or $\ell_1^2$) and the space whose unit sphere is an
equilateral hexagon. Finally, we address the question what are the spaces $E =
(\R^n, \|\cdot\|_E)$ with absolute norm such that for every collection $X_1,
\ldots, X_n$ of GL-spaces their $E$-sum is a GL-space.
