  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.
