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

TL;DR

This paper establishes sharp bounds for the affine invariant involving the volume of a convex body and its second projection body in three dimensions, with implications for isoperimetric inequalities and longstanding conjectures.

Contribution

It proves sharp bounds for the second projection body of 3D convex bodies, linking them to volume ratios and advancing understanding of affine invariants.

Findings

For 3D zonoids of volume 1, the second projection body is contained in 8 times the body.

For symmetric 3D convex bodies of volume 1, the second projection body contains 6 times the body.

Established a lower bound for the invariant P(K) in 3D and improved bounds for bodies of revolution in higher dimensions.

Abstract

We prove results relative to the problem of finding sharp bounds for the affine invariant $P(K)=V(\Pi K)/V^{d-1}(K)$. Namely, we prove that if $K$ is a 3-dimensional zonoid of volume 1, then its second projection body $\Pi^2K$ is contained in 8K, while if $K$ is any symmetric 3-dimensional convex body of volume 1, then $\Pi^2K$ 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.