It is shown that within the class of all multivariate distributions depending on a location parameter (and satisfying certain smoothness conditions) and with a weighted norm constraint on the covariance matrix, the one with minimum Fisherian information is the Gaussian distribution. This result is then used in obtaining a tight upper bound on the error of estimating an unknown random vector observed in additive Gaussian noise under quadratic loss.
"Optimum Fisherian Information for Multivariate Distributions." Ann. Statist. 5 (6) 1240 - 1244, November, 1977. https://doi.org/10.1214/aos/1176344009