Some Affine Invariants Revisited

TL;DR

This paper explores sharp inequalities and new definitions for affine invariants of convex bodies, revealing potential limits of normalized p-affine surface areas and their connections to centro-affine invariants.

Contribution

It introduces new inequalities for the SL(n) invariant and offers alternative definitions for the Paouris-Werner invariant, suggesting a unifying limit approach for affine invariants.

Findings

Established sharp inequalities for the SL(n) invariant _{2,n}(K)

Proposed two alternative definitions for the Paouris-Werner invariant _K

Conjectured that all SL(n) invariants with positive centro-affine curvature can be limits of normalized p-affine surface areas

Abstract

We present several sharp inequalities for the SL(n) invariant $\Omega_{2,n}(K)$ introduced in our earlier work on centro-affine invariants for smooth convex bodies containing the origin. A connection arose with the Paouris-Werner invariant $\Omega_K$ defined for convex bodies $K$ whose centroid is at the origin. We offer two alternative definitions for $\Omega_K$ when $K \in C^2_+$. The technique employed prompts us to conjecture that any SL(n) invariant of convex bodies with continuous and positive centro-affine curvature function can be obtained as a limit of normalized $p$-affine surface areas of the convex body.