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.

Geometric valuation theory

“…So-called Blaschke valuations were classified in [51]. For the corresponding question within the L p Brunn-Minkowski theory, see [77].…”

Complex-valued valuations on Lp spaces

“…Probably the most famous result on valuations is Hadwiger's classification theorem of continuous rigid motion invariant valuations. For the more recent contributions on valuations convex bodies can see [1,2,4,6,[12][13][14][15][16][17][18][19][20][21][22][23][24][26][27][28][29][30][31][32][33]38,39,[43][44][45].…”

Orlicz valuations

“…The first result on convex bodies valued valuations was obtained by Schneider [50] in the 1970s. During the last few decades, after a series of papers by Ludwig [25,[27][28][29], convex bodies valued valuations were studied quickly, see [11][12][13][14][15]18,19,24,32,[45][46][47][48][52][53][54][55][56] and also Ludwig's survey [30].…”