I would like to compare nutrient intake of men and women. But the assumption of the same variance-covariance matrix is violated and therefore two-sample Hotelling's $T^2$ test cannot be applied. How else can I compare the intakes?
P O'Brien has shown in http://www.citeulike.org/user/harrelfe/article/13264639 how to use the binary logistic model to compare a series of continuous and categorical variables. In your example, the principle is based on the fact that if there are no differences in intakes between men and women, you can't predict the sex of an individual by measuring their food intakes. The logistic approach makes far fewer assumptions than Hotelling's test.