  We prove interpolating estimates providing a bound for the oscillation of a
function in terms of two $L^p$ norms of its gradient. They are based on a
pointwise bound of a function on cones in terms of the Riesz potential of its
gradient. 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 for
Alexandrov's Soap Bubble Theorem and Serrin's overdetermined boundary value
problem. The new approach results in several novelties and benefits for these
problems.
