On the shape of a convex body with respect to its second projection body

We prove results relative to the problem of finding sharp bounds for the affine invariant $P(K)=V(\mathrm{\Pi }K)/V^{d-1}(K)$. Namely, we prove that if K is a 3-dimensional zonoid of volume 1, then its second projection body ${\mathrm{\Pi }}^{2}K$ is contained in 8K, while if K is any symmetric 3-dimensional convex body of volume 1, then ${\mathrm{\Pi }}^{2}K$ contains 6K. Both inclusions are sharp. Consequences of these results include a stronger version of a reverse isoperimetric inequality for 3-dimensional zonoids established by the author in a previous work, a reduction for the 3-dimensional Petty conjecture to another isoperimetric problem and the best known lower bound up to date for $P(K)$ in 3 dimensions. As byproduct of our methods, we establish an almost optimal lower bound for high-dimensional bodies of revolution.