About the optimal density associated to the chiral index of a sample from a bivariate distribution

The complex quadratic form z′ Pz, where z is a fixed vector in Cn and z′ is its transpose, and P is any permutation matrix, is shown to be a convex combination of the quadratic forms z′ Pσz, where Pσ denotes the symmetric permutation matrices. We deduce that the optimal probability density associated to the chiral index of a sample from a bivariate distribution is symmetric. This result is used to locate the upper bound of the chiral index of any bivariate distribution in the interval [1 - 1/π, 1 - 1/2π]. © 2005 Académie de sciences. Published by Elsevier SAS. All rights reserved.