Here I describe the solution to the problem from the last post. Spoilers!
As one may take the simplex Personally, I think about it as follows: start at the origin, then go to the point then to etc. Finally take the convex hull of all the vertices to obtain the simplex. The intuition is that each time forming a new vertex, we go out of the subspace spanned by all previous ones.
Obviously, On the other hand, It is easy to obtain by some low-level convex optimization reasoning (it’s a good exercise though!) that and in fact it is attained at any vertex.