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arXiv · 1804.07015

Double normals of most convex bodies

Alain Rivière·Joël Rouyer·Costin Vîlcu·Tudor Zamfirescu
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Abstract

We consider a typical (in the sense of Baire categories) convex body $K$ in $\mathbb{R}^{d+1}$. The set of feet of its double normals is a Cantor set, having lower box-counting dimension $0$ and packing dimension $d$. The set of lengths of those double normals is also a Cantor set of lower box-counting dimension $0$. Its packing dimension is equal to $\frac{1}{2}$ if $d=1$, is at least $\frac{3}{4}$ if $d=2$, and equals $1$ if $d\geq3$. We also consider the lower and upper curvatures at feet of double normals of $K$, with a special interest for local maxima of the length function (they are countable and dense in the set of double normals). In particular, we improve a previous result about the metric diameter.

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Alain Rivière, Joël Rouyer, Costin Vîlcu, Tudor Zamfirescu. 2018-04-19. Double normals of most convex bodies. https://arxiv.org/abs/1804.07015

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