A remarkably simple color constancy method was recently developed, based in essence on the Gray-Edge method, i.e., the assumption that the mean of color-gradients in a scene (or colors themselves, in a Gray-World setting) are close to being achromatic. However this new method for illuminant estimation explicitly includes the important notions that (1) we cannot hope to recover illuminant strength, but only chromaticity; and (2) that a polynomial regression from image moment vectors to chromaticity triples should be based not on polynomials but instead on the roots of polynomials, in order to release the regression from absolute units of lighting. In this paper we extend these new image moments in several ways: by replacing the standard expectation value mean used in the moments by a Minkowski p-norm; by going over to a float value for the parameter p and carrying out a nonlinear optimization on this parameter; by considering a different expectation value, generated by using the geometric mean. We show that these strategies can drive down the median and maximum error of illumination estimates.