Abstract

An old result of Zamfirescu says that for most convex curves C in the plane most points in R2\mathbb{R}^2 lie on infinitely many normals to C, where most is meant in Baire category sense. We strengthen this result by showing that 'infinitely many' can be replaced by 'continuum many' in the statement. We present further theorems in the same spirit.

Author information

Contact details are reproduced from the original publication and may be historical.

Imre Bárány

A. Rényi Mathematical Institute, Hungarian Academy of Sciences, PO Box 127, 1364 Budapest, Hungary
and: Department of Mathematics, University College London, Gower Street, London, WC1E 6BT, UK

barany@renyi.hu

Miklós Laczkovich

Dept. of Analysis, Eötvös University, Pázány Péter Sétány 1/c, 1117 Budapest, Hungary
and: Department of Mathematics, University College London, Gower Street, London, WC1E 6BT, UK

laczk@cs.elte.hu

Keywords

  • Convex curves in the plane
  • affine diameters
  • Baire category

Mathematics Subject Classification

52A10; 54E52

References

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Suggested citation

I. Bárány, M. Laczkovich. “Magic Mirrors, Dense Diameters, Baire Category.” Journal of Convex Analysis 24 (2017), No. 1, 93–102. https://doi.org/10.68381/jca24007

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Published by Heldermann Verlag, 2017. Rights now held by Banach Press LLC.