Conditions equivalent to central symmetry of convex curves

Abstract

In this paper we establish that a convex planar body C is centrally symmetric provided either one of the following conditions hold: (1) Each line halving the circumference of the boundary γ of C is a diametral line.(A diametral line is a line intersecting C in a chord of maximal length in the family of parallel chords.) (2) Each line halving the area of C is a diametral line.