$L_p$-Steiner quermassintegrals

TL;DR

This paper introduces $L_p$-Steiner quermassintegrals, generalizing classical geometric measures, and explores their properties, invariances, and potential novelty within convex geometry.

Contribution

It defines $L_p$-Steiner quermassintegrals inspired by an $L_p$ Steiner formula and investigates their properties, invariances, and novelty as new valuations.

Findings

$L_p$-Steiner quermassintegrals generalize classical measures.

They are rotation and reflection invariant valuations.

These valuations are potentially new in convex geometry.

Abstract

Inspired by an $L_p$ Steiner formula for the $L_p$ affine surface area proved by Tatarko and Werner, we define, in analogy to the classical Steiner formula, $L_p$-Steiner quermassintegrals. Special cases include the classical mixed volumes, the dual mixed volumes, the $L_p$ affine surface areas and the mixed $L_p$ affine surface areas. We investigate the properties of the $L_p$-Steiner quermassintegrals in a special class of convex bodies. In particular, we show that they are rotation and reflection invariant valuations in this class of convex bodies with a certain degree of homogeneity. Such valuations seem new and have not been observed before.