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Let $K\subset\R^3$ be a convex body and let $\Pi K$ be its projection body. We prove \[ \frac{V_3(\Pi K)}{V_3(K)^2} \ge \frac{3\pi^2}{4}, \] with equality exactly for ellipsoids. The proof combines a section estimate of Saroglou with a sharp one-dimensional functional inequality. The latter is proved by passing to the distribution function and approximating it by step functions. We also prove the complementary sharp upper bound \( V_3(\Pi K)/V_3(K)^2\le18, \) which is attained by tetrahedra.

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