Symmetric decreasing rearrangement (when applicable) can reduce a geometric variational problem to a radial problem, where the unknown functions depend on the single variable |x|. Classical inequalities for perimeter, gradient norms, and convolution integrals indicate that symmetric decreasing rearrangement reduces the overall oscillation of functions. Less is known about its effect on the mean oscillation of a function. I will discuss recent result (w. Galia Dafni and Ryan Gibara) on inequalities and continuity properties. The question of sharp inequalities remains open.