The natural projection of your exterior point $a$ is to the point $b$ on the polytope that is
closest to $a$, i.e., which minimizes the distance $|ab|$.  This can be formulated as a [quadratic
programming problem][1], for which there are many algorithms.  

Quite some time ago, Gilbert worked out some methods:

> (1) E. G. Gilbert,  "Minimizing the quadratic form on a convex set",  *SIAM J. Contr.*,  vol. 4,  pp.61-79 1966 

> (2) E. G. Gilbert, D. W. Johnson, and S. S. Keerthi,  "A fast procedure for computing the distance between complex objects in three dimensional space",  *IEEE J. Robot. Automat.*,  vol. 4,  pp.193-203 1988 ([PDF link][2])

The first sentence of the 2nd paper above is: "An efficient and reliable algorithm for computing
the Euclidean distance between a pair of convex sets in $\mathbb{R}^m$ is described."  This algorithm has become known as the *GJK algorithm*.

I doubt this is the last word on the topic.  There is a huge literature on collision detection in
$\mathbb{R}^3$—which often amounts to finding the minimum distance from a point to a polyhedron—but I don't know how much of it scales gracefully to dimensions $5$ or $6$. 


  [1]: http://en.wikipedia.org/wiki/Quadratic_programming
  [2]: http://graphics.stanford.edu/courses/cs448b-00-winter/papers/gilbert.pdf