Concurrent normals to convex bodies and spaces of Morse functions

It is conjectured K ⊂ ℝ ⁿ is a convex body, then there exists a point in the interior of K which is the point of concurrency of normals from 2n points on the boundary of K. We present a topological proof of this conjecture in dimension four assuming ∂K is C ^1,1. From the assumption that the conjecture fails for K ⊂ ℝ ⁴, we construct a retraction from K̄ to ∂K. We apply the same strategy to the problem for lower n, assuming no regularity on ∂K, and show that it provides very simple proofs for the cases of two and three dimensions (the dimension three case was first proved by Erhard Heil). A connection between our approach to this problem and the homotopy type of some function spaces is also explored, and some conjectures along those lines are proposed.