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We prove interpolating estimates providing a bound for the oscillation of afunction in terms of two $L^p$ norms of its gradient. They are based on apointwise bound of a function on cones in terms of the Riesz potential of itsgradient. The estimates hold for a general class of domains, including, e.g.,Lipschitz domains. All the constants involved can be explicitly computed.  As an application, we show how to use these estimates to obtain stability forAlexandrov's Soap Bubble Theorem and Serrin's overdetermined boundary valueproblem. The new approach results in several novelties and benefits for theseproblems.

Interpolating estimates with applications to some quantitative symmetry results