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We show that if $K \\supseteq r B_2^n$ then: \\[ \\sqrt{n} M(K) \\leqslant C \\sum_{k=1}^{n} \\frac{1}{\\sqrt{k}} \\min\\left(\\frac{1}{r} , \\frac{n}{k} \\log\\Big(e + \\frac{n}{k}\\Big) \\frac{1}{v_{k}^{-}(K)}\\right) . \\] where $M(K)=\\int_{S^{n-1}} \\|x\\|\\, d\\sigma(x)$ is the mean-norm, $C>0$ is a universal constant, and $v^{-}_k(K)$ denotes the minimal volume-radius of a $k$-dimensional orthogonal projection of $K$. 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We show that if $K \\supseteq r B_2^n$ then: \\[ \\sqrt{n} M(K) \\leqslant C \\sum_{k=1}^{n} \\frac{1}{\\sqrt{k}} \\min\\left(\\frac{1}{r} , \\frac{n}{k} \\log\\Big(e + \\frac{n}{k}\\Big) \\frac{1}{v_{k}^{-}(K)}\\right) . \\] where $M(K)=\\int_{S^{n-1}} \\|x\\|\\, d\\sigma(x)$ is the mean-norm, $C>0$ is a universal constant, and $v^{-}_k(K)$ denotes the minimal volume-radius of a $k$-dimensional orthogonal projection of $K$. 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