# An optimal inequality on locally strongly convex centroaffine hypersurfaces

## TL;DR

This paper establishes an optimal inequality for locally strongly convex centroaffine hypersurfaces in Euclidean space, relating covariant derivatives of key geometric tensors, and classifies the hypersurfaces achieving equality.

## Contribution

It introduces a new optimal inequality involving the difference tensor and Tchebychev vector field, with a complete classification of equality cases.

## Key findings

- Derived a general inequality for convex hypersurfaces
- Classified hypersurfaces that attain equality in the inequality
- Connected the inequality to hypersurfaces with parallel cubic form

## Abstract

In this paper, we establish a general inequality for locally strongly convex centroaffine hypersurfaces in $\mathbb{R}^{n+1}$ involving the norm of the covariant derivatives of both the difference tensor $K$ and the Tchebychev vector field $T$. Our result is optimal in that, applying our recent classification for locally strongly convex centroaffine hypersurfaces with parallel cubic form in [4], we can completely classify the hypersurfaces which realize the equality case of the inequality.