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.