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Given a risk position X, a random addition Z is called a risk reducer for X if the new position X+ Z is less risky than X+ E [Z] in convex order. We utilize the concept of convex hull to give a structural description of risk reducers in the case of an atomless probability space. Then we study risk reducers that are fully dependent on X. Applications to multivariate stochastic ordering, index-linked hedging strategies, and optimal reinsurance are proposed.