(07/06/2009, 12:53 AM)BenStandeven Wrote: the path would have to pass through a point with imaginary value 2 pi, and also through its conjugate. Then the other side of the region would intersect itself at the exponential of that point.

Say the curve is injective and connects two points and with equal real part and with . One needs to show that then there is always a pair of points and with equal real part and with .

This sounds very plausible but I couldnt prove it except for certain simple shapes of , e.g. convex.