Based on the study of geometric properties of grand Lebesque space, we give an exact description of the mutual embeddings of grand Lebesgue spaces \(\overrightarrow{L^{\alpha _0}}(\zeta )\) and classical Orlicz Lorentz and Marcinkiewicz spaces. It turns out that the grand Lebesgue space \(\overrightarrow{L^{\alpha _0}}(\zeta )\) for \(\alpha _0\in (0, 1)\) cannot be described as an Orlicz, Lorentz, and Marcinkiewicz space. We also show that the grand Lebesgue space \(\overrightarrow{L^{\alpha _0}}(\zeta )\) for \(\alpha _0\in [0, 1)\) is not reflexive. Using duality, we give similar results for small Lebesgue spaces \(\overleftarrow{L^{\alpha _1}}(\xi )\) .
