Valuations on the space of quasi-concave functions

TL;DR

This paper characterizes valuations on quasi-concave functions in Euclidean space, focusing on properties like invariance, continuity, and monotonicity, providing a comprehensive mathematical framework.

Contribution

It offers a complete description of valuations on quasi-concave functions that are invariant, continuous, and monotone, advancing the theoretical understanding of these function spaces.

Findings

Characterization of valuations invariant under rigid motions

Description of continuous valuations on quasi-concave functions

Identification of conditions for monotone valuations

Abstract

We characterize the valuations on the space of quasi-concave functions defined on the $N$-dimensional Euclidean space, 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.