There is a nice paper on a similar topic by [Burger, Gritzmann and Klee "Polytope projection and projection polytopes"][1] .  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. 


  [1]: http://www.jstor.org/pss/2974444