Central Symmetry of Star-Shaped Flat Bodies

Abstract

Well-known criteria for the central symmetry are formulated for convex bodies. This study relates to a broader class of star-shaped bodies but is limited by the dimension of 2. The paper introduces the concepts of a sector and a segment of a flat star-shaped body.The basic result is the following. Let a flat body K be star-shaped with respect to its interior point o. On the set of sectors and segments of K, a simply additive, monotonic, and invariant with respect to central symmetry with the center o functional F is given. The body K is centrally symmetric with respect to the center o if and only if every chord passing through the point o divides K into two sectors with equal values of the functional F.The method of proof is — "on the contrary".When considering quantities having geometric meaning (central geometric moments, area) as such functionals, we get both new and known (for an area) statements for flat convex bodies. A slight modification of the proof allows us to obtain a similar statement for the perimeter (an additive functional, but simply not an additive functional on the set of convex flat bodies): flat convex body has its central symmetry if and only if all the chords, dividing the perimeter into halves, pass through one point.