, Volume 352, Issue 1, pp 55-71

Concurrent normals to convex bodies and spaces of Morse functions

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Abstract

It is conjectured that if ${K\subset\mathbb R^n}$ 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 ${\partial K}$ is C 1,1. From the assumption that the conjecture fails for ${K\subset\mathbb R^4}$ , we construct a retraction from ${\overline K}$ to ${\partial K}$ . We apply the same strategy to the problem for lower n, assuming no regularity on ${\partial K}$ , 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.