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Journal of Convex Analysis 24 (2017), No. 1, 093--102
Copyright Heldermann Verlag 2017

Magic Mirrors, Dense Diameters, Baire Category

Imre Bárány
A. Rényi Mathematical Institute, Hungarian Academy of Sciences, PO Box 127, 1364 Budapest, Hungary

Miklós Laczkovich
Dept. of Analysis, Eötvös University, Pázány Péter Sétány 1/c, 1117 Budapest, Hungary

An old result of Zamfirescu says that for most convex curves C in the plane most points in R2 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.

Keywords: Convex curves in the plane, affine diameters, Baire category.

MSC: 52A10; 54E52

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