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A note on the concurrent normal conjecture

Abstract

It is conjectured since long that for any convex body \(K \in \mathbb{R}^n\) there exists a point in the interior of K which belongs to at least 2n normals from different points on the boundary of K. The conjecture is known to be true for n = 2, 3, 4.

Motivated by a recent preprint of Y. Martinez-Maure [4], we give a short proof of his result: for dimension \(n\geq 3\), under mild conditions, almost every normal through a boundary point to a smooth convex body \(K \in \mathbb{R}^n\) contains an intersection point of at least 6 normals from different points on the boundary of K.

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References

  1. V. Arnol'd, A. Varchenko and S. Gusein-Zade, Singularities of Differentiable Maps, I, II, Birkhäuser (1985, 1988).

  2. E. Heil,Concurrent normals and critical points under weak smoothness assumptions, in: Discrete Geometry and Convexity (New York, 1982), Ann. New York Acad. Sci., vol. 440, New York Acad. Sci. (New York, 1985), pp. 170–178.

  3. E. Heil, Existenz eines 6-Normalenpunktes in einem konvexen Körper, Arch. Math. (Basel), 32 (1979), 412–416.

  4. Y. Martinez-Maure, On the concurrent normals conjecture for convex bodies, hal-03292275v3 (2021).

  5. J. Milnor, Lectures on the h-Cobordism Theorem, Princeton University Press (Princeton, NJ, 1965).

  6. J. Pardon, Concurrent normals to convex bodies and spaces of Morse functions, Math. Ann., 352 (2012), 55–71.

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Correspondence to G. Panina.

Additional information

A. Grebennikov is supported by Ministry of Science and Higher Education of the Russian Federation, agreement 075-15-2019-1619.

G. Panina is supported by RFBR grant 20-01-00070A.

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Grebennikov, A., Panina, G. A note on the concurrent normal conjecture. Acta Math. Hungar. 167, 529–532 (2022). https://doi.org/10.1007/s10474-022-01251-0

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  • DOI: https://doi.org/10.1007/s10474-022-01251-0

Key words and phrases

  • bifurcation
  • Morse–Cerf theory
  • Morse point

Mathematics Subject Classification

  • 52A07