On scalar invariant maps and applications

Scalar invariance is a fundamental property in the theory of risk measures. In this paper, we investigate scalar invariant maps on $ {L^{\infty}} $ through two main directions. First, we introduce the notions of scalar closedness and scalar solidity for pseudo-acceptance sets and show that these properties provide a necessary and sufficient characterization of acceptance sets without requiring convexity or $ {\mathrm{weak}}^*{\text{-closedness}} $. Second, we develop a generalized duality framework between $ {L^{\infty}} $ and the class of scalar invariant maps, based on the evaluation pairing $ (h,\phi)\rightarrow \phi(h) $. This leads to a bipolar-type representation for acceptance sets under minimal structural assumptions. Finally, we introduce a hull operator associated with abstract properties on scalar invariant maps, interpret it as the largest minorant satisfying a given property, and study consistency and representation results. Several classical properties (convexity, subadditivity, monotonicity, etc.) are revisited within this framework, and explicit formulas for the associated hulls are provided.