Random normed spaces from isotropic log-concave measures satisfy d_BM >= cn / ln(1+m/n) with high probability, sharp in both parameters and recovering the order-n extremal when m is linear in n.
Tomczak-Jaegermann,Banach–Mazur Distances and Finite Dimensional Operator Ideals, Pitman Mono- graphs and Surveys in Pure and Applied Mathematics, 38
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Banach-Mazur distances and basis constants of isotropic log-concave random spaces
Random normed spaces from isotropic log-concave measures satisfy d_BM >= cn / ln(1+m/n) with high probability, sharp in both parameters and recovering the order-n extremal when m is linear in n.