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This paper presents novel characterization results for classes oflaw-invariant star-shaped functionals. We begin by establishingcharacterizations for positively homogeneous and star-shaped functionals thatexhibit second- or convex-order stochastic dominance consistency. Building onthese characterizations, we proceed to derive Kusuoka-type representations forthese functionals, shedding light on their mathematical structure and intimateconnections to Value-at-Risk and Expected Shortfall. Furthermore, we offerrepresentations of general law-invariant star-shaped functionals asrobustifications of Value-at-Risk. Notably, our results are versatile,accommodating settings that may, or may not, involve monotonicity and/orcash-additivity. All of these characterizations are developed within a generallocally convex topological space of random variables, ensuring the broadapplicability of our results in various financial, insurance and probabilisticcontexts.

Law-Invariant Return and Star-Shaped Risk Measures