Symmetrization of convex plane curves

Several point symmetrizations of a convex curve Gamma are introduced and one, the affinely invariant 'central symmetric transform' (CST) with respect to a given basepoint inside Gamma, is investigated in detail. Examples for Gamma include triangles, rounded triangles, ellipses, curves defined by support functions and piecewise smooth curves. Of particular interest is the region of basepoints for which the CST is convex (this region can be empty but its complement in the interior of Gamma is never empty). The (local) boundary of this region can have cusps and in principle it can be determined from a geometrical construction for the tangent direction to the CST.