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In this paper, we put forth a novel framework (named ``RYU'') for theconstruction of ``safe'' balls, i.e. regions that provably contain the dualsolution of a target optimization problem. We concentrate on the standard setupwhere the cost function is the sum of two terms: a closed, proper, convexLipschitz-smooth function and a closed, proper, convex function. The RYUframework is shown to generalize or improve upon all the results proposed inthe last decade for the considered family of optimization problems.

One to beat them all: "RYU'' -- a unifying framework for the construction of safe balls