I’ve developed a method to compute the closest point to the origin on any simplex (triangle, tetrahedron, etc.) of any degree, in any dimension. This unified algorithm also allows just one function to be applied to any type of simplex, as opposed to separate algorithms for different types used by earlier methods. On it’s own this might not sound very exciting, but this allows the common GJK algorithm to operate on convex shapes embedded in any dimension.
A paper detailing the design of this algorithm may be found here, and a reference implementation may be found here.