Pure and Applied Mathematics Quarterly
Volume 16 (2020)
Symmetrization of convex plane curves
Pages: 1767 – 1787
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.
affine invariants, convex plane curves, singularities, symmetrization, higher inflections
2010 Mathematics Subject Classification
Primary 52A10. Secondary 53A04.
Received 1 May 2020
Published 17 February 2021