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Set-valued quantiles for multivariate distributions with respect to a generalconvex cone are introduced which are based on a family of (univariate)distribution functions rather than on the joint distribution function. It isshown that these quantiles enjoy basically all the properties of univariatequantile functions. Relationships to families of univariate quantile functionsand to depth functions are discussed. Finally, a corresponding Value at Riskfor multivariate random variables as well as stochastic orders are introducedvia the set-valued approach.

Cone distribution functions and quantiles for multivariate random variables