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$$\\mu(K)\\,\\leq\\,\\frac{n}{n-k}c_{n,k}\\max_{H} \\mu(K\\cap H) \\vol_n(K)^{k/n}.$$ Here $c_{n,k} = |B_2^n|^{(n-k)/n}/|B_2^{n-k}|<1,$ $|B_2^n|$ is the volume of the unit Euclidean ball, and maximum is taken over all $(n-k)$-dimensional subspaces of $\\R^n.$ The constant is optimal, and for each intersection body the inequality 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