𝐿_{𝑝}-Blaschke valuations

Abstract

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