Concurrent normals to convex bodies and spaces of Morse functions |
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Authors: | Email author" target="_blank">John?PardonEmail author |
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Institution: | 1.Department of Mathematics,Princeton University,Princeton,USA |
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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. |
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