$L_p$-Blaschke Valuations

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 $n\ge 3$, 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 $p\ne n$ are also discussed by studying the relations between symmetric $L_p$-Blaschke valuations and normalized ones.

Cited by in corpus (2)

• Laplace transforms and valuations
• Real-valued valuations on Sobolev spaces