There is a nice paper on a similar topic by Burger, Gritzmann and Klee "Polytope projection and projection polytopes" . They describe an algorithm to compute the minimum surface area projection of an n-dimensional simplex. Accord to the paper it is NP-hard to find the maximum surface area of a n dimensional simplex.

There is a nice paper on a similar topic by Burger, Gritzmann and Klee "Polytope projection and projection polytopes" . They describe an $O(n^2)$ algorithm to compute the minimum surface area projection of an n-dimensional simplex. According to the paper it is NP-hard to find the maximum surface area projection of a n-dimensional simplex.