  We show that the boundary of a rotating vortex patch (or V-state, in the
terminology of Deem and Zabusky) is of class C^infinity provided the patch is
close enough to the bifurcation circle in the Lipschitz norm. The rotating
patch is convex if it is close enough to the bifurcation circle in the C^2
norm. Our proof is based on Burbea's approach to V-states. Thus conformal
mapping plays a relevant role as well as estimating, on H\"older spaces,
certain non-convolution singular integral operators of Calder\'on-Zygmund type.
