On $L_p$ Affine Surface Area and Curvature Measures

TL;DR

This paper explores the connection between $L_p$ affine surface area and curvature measures, introducing a new curvature-based representation and proving its equivalence to existing notions, highlighting dualities with prior work.

Contribution

It provides a novel curvature-based representation of $L_p$ affine surface area and establishes its equivalence to previous definitions through direct proofs.

Findings

New curvature measure representation of $L_p$ affine surface area

Proof of equivalence with existing definitions

Identification of dualities with prior formulations

Abstract

The relationship between $L_p$ affine surface area and curvature measures is investigated. As a result, a new representation of the existing notion of $L_p$ affine surface area depending only on curvature measures is derived. Direct proofs of the equivalence between this new representation and those previously known are provided. The proofs show that the new representation is, in a sense, "polar" to that of Lutwak's and "dual" to that of Sch\"utt & Werner's.