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For every gauge $\\phi$ and every $q\\geqslant 1$, $$ \\|\\phi(G)\\|_q\\leqslant C\\sqrt{\\ln(en)+q}\\,\\|\\phi(X)\\|_q,\\qquad \\|\\phi(X)\\|_q\\leqslant C\\left(\\sqrt{\\ln(en)}+\\psi(X)\\sqrt q\\right)\\|\\phi(G)\\|_q. $$ Applied to support functions, this gives the sharp worst case order $C\\sqrt{\\ln(en)}$ for the $L_2$-Sudakov constant and quantitative $L_p$-Sudakov estimates. 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