Valuations on the Space of Quasi-Concave Functions

Abstract

We characterize the valuations on the space of quasi-concave functions on R N , that are rigid motion invariant and continuous with respect to a suitable topology. Among them we also provide a specific description of those which are additionally monotone.

A homogeneous decomposition theorem for valuations on convex functions

“…Valuations on function spaces have only recently started to attract attention. Classification results were obtained for L p and Sobolev spaces [24-27, 29, 38, 39], spaces of quasi-convex functions [12,13], of Lipschitz functions [17], of definable functions [4] and on Banach lattices [37]. Spaces of convex functions play a special role because of their close connection to convex bodies.…”

Translation invariant valuations on quasi-concave functions

“…Under suitable assumptions on φ, the functional µ is a well-defined, continuous and rigid motion invariant valuation. Indeed, we have the following result (for the proof see [4]). and the condition on φ is not necessary.…”

“…The function c is continuous by the continuity of µ and it is univocally determined by (13). The integral form of µ and the additional condition on c can be obtained using the same argument of the proof of Lemma 6.5 and Theorem 1.2 of [4], that we briefly outline. We first consider a general simple function f of the form…”

“…The results achieved so far embrace various types of spaces, such as Lebesgue spaces, Sobolev spaces, functions of bounded variations, convex functions, etc. For more details the reader is referred to the papers: [[2]- [4], [9]- [16], [18], [21]- [25]].…”

“…In the paper [4], the first two authors considered valuations µ on C N which are invariant with respect to the composition with rigid motions, i.e. µ(f • T ) = µ(f ) for every f ∈ C N and for every isometry T (i.e.…”