Generalized-lush spaces revisited

Abstract

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 $E=({\mathbb{R}}^n,\Vert \cdot {\Vert }_E)$ with absolute norm such that for every collection $X_1,\dots ,X_n$ of GL-spaces their $E$-sum is a GL-space.