S Normal distributions as maximum-entropy distributions. Let C be a positive definite symmetric nn matrix, and consider the class WC of all probability measures P on .Rn;Bn/

with the properties F P is centred with covariance matrix C, that is, the projections Xi W Rn ! R satisfy E.Xi / D 0 and Cov.Xi;Xj / D Cij for all 1  i; j  n, and F P has a Lebesgue density , and there exists the differential entropy H.P/ D 

Z Rn dx .x/ log .x/ :

Show that H.Nn.0; C// D n

2 log 2e.det C/1=n D max P2WC H.P/ :

Hint: Consider the relative entropy H.PINn.0; C//; cf. Remark (7.31).