# Generalized-lush spaces revisited

## TL;DR

This paper explores geometric properties of GL-spaces, showing finite-dimensional cases are polyhedral, classifying 2D GL-spaces, and investigating conditions for their sums to remain GL-spaces.

## Contribution

It provides a classification of finite-dimensional GL-spaces and characterizes 2D GL-spaces, advancing understanding of their geometric structure and sum properties.

## Key findings

- Finite-dimensional GL-spaces are polyhedral.
- In dimension 2, only two GL-spaces exist up to isometry.
- Conditions identified for E-sums of GL-spaces to be GL-spaces.

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