.. Best wishes! Exactly where are your contact

details though? ]]>

Is there any proof using Fourier analysis for the n-dimensional case? ]]>

Cabre, X., “Elliptic PDE’s in probabilities in geometry and symmetry and regularity of solutions,” Disc. Cont. Dynamical Systems 20 (3), 425-457 (2008).

Don’t be turned off by the title if you are scared of PDEs, it is elementary.

]]>For a region bounded by a (piecewise) smooth curve, we can see again that it must be convex (by the same argument as for the polygons). To prove that the boundary must be a circle, we inscribe any equilateral polygon into it and observe that it must be regular, otherwise the area of the region can be increased by making this polygon regular by changing its angles and keeping the segments of the region adjacent to the sides of the polygon the same, because the area of the whole region is the area of the inscribed polygon + the total area of the segments cut off from this region by the sides of the polygon. Q.E.D.

]]>