Reflections of planar convex bodies

TL;DR

This paper proves that in the plane, every convex body has a special point where reflecting the body results in a convex union, and non-centrally symmetric bodies have three such affinely independent points.

Contribution

The paper establishes the existence of a reflection point for convex bodies in the plane and identifies three affinely independent points for non-symmetric bodies.

Findings

Existence of a reflection point with convex union for all convex bodies.

Non-centrally symmetric bodies have three affinely independent points with this property.

The results apply specifically to planar convex bodies.

Abstract

It is proved that every convex body in the plane has a point such that the union of the body and its image under reflection in the point is convex. If the body is not centrally symmetric, then it has, in fact, three affinely independent points with this property.