  One may define a trilinear convolution form on the sphere involving two
functions on the sphere and a monotonic function on the interval $[-1,1]$. A
symmetrization inequality of Baernstein and Taylor states that this form is
maximized when the two functions on the sphere are replaced with their
nondecreasing symmetric rearrangements. In the case of indicator functions, we
show that under natural hypotheses, the symmetric rearrangements are the only
maximizers up to symmetry by establishing a sharpened inequality.
