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This result improves the former best known lower bound, of order $\\sqrt{\\frac{n}{\\log \\log(n)}}$.\n  We also study the exact asymptotic behavior of the largest volume ratio for some natural classes. 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This result improves the former best known lower bound, of order $\\sqrt{\\frac{n}{\\log \\log(n)}}$.\n  We also study the exact asymptotic behavior of the largest volume ratio for some natural classes. 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