Translation invariant valuations on quasi-concave functions

Abstract

Abstract. We study real-valued, continuous and translation invariant valuations defined on the space of quasi-concave functions of N variables. In particular, we prove a homogeneous decomposition theorem of McMullen type, and we find a representation formula for those valuations which are N -homogeneous. Moreover, we introduce the notion of Klain's functions for this type of valuations.

A homogeneous decomposition theorem for valuations on convex functions

“…We prove, by contradiction, that ζ has compact support. Assume that there exists a sequence y k ∈ R n , such that (13) lim k→∞ |y k | = +∞ and ζ(y k ) = 0 for every k. Without loss of generality, we may assume that…”

Geometric valuation theory

“…See [40], for more information on homogeneous decompositions and why such results do not hold for many spaces of convex functions. For more results on valuations on convex functions, see [15,34,68,69], and for results on valuations on quasi-concave functions, see [35,36]. While formally not results for valuations on function spaces, classification results for valuations on star shaped sets in R n were the motivation for some of the results on function spaces.…”

Smooth valuations on convex functions

“…In recent years, valuations on a variety of well known classes of functions have been studied and classified, including Sobolev-spaces [27,28,30], L p -spaces [29,35,39,40], quasi-concave functions [8,11,12], Orlicz-spaces [26], functions of bounded variation [41], and convex functions [5,9,13,14,15,16,25,33,34].…”