## Abstract

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.

Original language | English (US) |
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Pages (from-to) | 55-71 |

Number of pages | 17 |

Journal | Mathematische Annalen |

Volume | 352 |

Issue number | 1 |

DOIs | |

State | Published - Jan 1 2012 |

## All Science Journal Classification (ASJC) codes

- Mathematics(all)