### Abstract or Additional Information

Two-point symmetrizations are simple rearrangements that have been used to prove isoperimetric inequalities on the sphere. For each unit vector u, there is a two-point symmetrization that pushes mass towards u across the normal hyperplane. A key point is that the only sets invariant under all two-point symmetrizations are the entire sphere and the empty set; if the directions u are restricted to the positive hemisphere, then the polar caps are invariant as well.

I will discuss work with Greg Chambers and Anne Dranovski, in the context of recent and classical symmetrization results.

How can full rotational symmetry be recovered from partial information? It is known that the reflections at d hyperplanes in general position generate a dense subgroup of O(d); in particular, a continuous function that is symmetric under these reflections must be radial. How many two-point symmetrizations are needed to verify that a function which increases under these symmetrizations is radial? I will show that d+1 such symmetrizations suffice, and will discuss the ergodicity of the random walk generated by the corresponding folding maps on the sphere.