Non-parametric estimation of smooth functions belonging to Sobolev classes is closely related to the problem of estimating the (infinite dimensional) mean of a standard Gaussian shift when the mean is known to lie in an body. Donoho, Liu and MacGibbon (1990) obtained exact results for the comparison between the linear maximum risk and the minimax risk among all estimates for this problem with Gaussian white noise. Here we study minimax estimation of infinite dimensional exponential family mean parameters constrained to belong to bodies under different loss functions. The results are illustrated using the Poisson distribution.