Mathematische Annalen

, Volume 352, Issue 1, pp 55–71 | Cite as

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 C1,1. 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.

Mathematics Subject Classification (2000)

52A20 54H25 

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© Springer-Verlag 2010

Authors and Affiliations

  1. 1.Department of MathematicsPrinceton UniversityPrincetonUSA

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