$L_p$-Blaschke Valuations

TL;DR

This paper classifies continuous, symmetric $L_p$-Blaschke valuations, extending Haberl's work, and identifies the normalized $L_p$-curvature image operator as the unique valuation in higher dimensions.

Contribution

It extends the classification of symmetric $L_p$-Blaschke valuations to all dimensions, including the case $p=n$, and introduces normalized valuations for better analysis.

Findings

For $n \\geq 3$, the normalized $L_p$-curvature image operator is unique.

In dimension 2, a rotated version is also an additional valuation.

The approach handles the case $p=n$ effectively and discusses relations for $p \\neq n$.

Abstract

In this article, a classification of continuous, linearly intertwining, symmetric $L_p$-Blaschke ($p>1$) valuations is established as an extension of Haberl's work on Blaschke valuations. More precisely, we show that for dimensions $n\geq 3$, the only continuous, linearly intertwining, normalized symmetric $L_p$-Blaschke valuation is the normalized $L_p$-curvature image operator, while for dimension $n = 2 $, a rotated normalized $L_p$-curvature image operator is an only additional one. One of the advantages of our approach is that we deal with normalized symmetric $L_p$-Blaschke valuations, which makes it possible to handle the case $p=n$. The cases where $p \not =n$ are also discussed by studying the relations between symmetric $L_p$-Blaschke valuations and normalized ones.