The middle hedgehog of a planar convex body

Abstract

A convexity point of a convex body is a point with the property that the union of the body and its reflection in the point is convex. It is proved that in the plane a typical convex body (in the sense of Baire category) has infinitely many convexity points. The proof makes use of the `middle hedgehog' of a planar convex body $K$, which is the curve formed by the midpoints of all affine diameters of $K$. The stated result follows from the fact that for a typical planar convex body the convex hull of the middle hedgehog has infinitely many exposed points.