Let $v_0, \dots, v_n \in R^n$ be $n+1$ affinely independent points. Let $\sigma = \text{conv}(v_0, \dots, v_n) \subset R^n$ be a general simplex.
I'm interested in a fast method for computing the orthogonal projection $$ \arg \min_{u \in \sigma} ~ \|u-v\|. $$
I know that for the probability simplex there exist efficient methods based on sorting. However I feel they do not extend to the general case.