GEOMETRIC-PROPERTIES OF CHIRAL BODIES

The new developments concerning the possible metrization of structural chirality have drawn much attention recently. The main approach of such quantification is based on the maximal volume overlap between two enantiomorphs of a given chiral set. This approach raises an interesting problem concerning the shape of such a domain of overlap, namely, whether it is chiral or not. It is shown presently that for a two or three dimensional set if the maximal volume overlap is unique then it must be achiral. It is also shown that if a two-dimensional body is convex then by the Brunn-Minkowski theorem the maximal volume overlap of the body with its enantiomorph is achiral. In addition, universal upper bounds for chiral coefficients chi(n) of convex sets in any dimension n are given, being chi(2) less than or equal to 0.3954 and chi(3) less than or equal to 0.6977 for dimensions two and three, respectively.