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Double normals of most convex bodies. (English) Zbl 1421.52005

A classical well-known result establishes that every normal to a convex body \(K\) (chord orthogonal to some supporting hyperplane of \(K\) at one of the end-points) is a double normal if and only if \(K\) has constant width. Moreover, in [Israel J. Math. 2, 71–80 (1964; Zbl 0131.37702)] N. H. Kuiper proved that every convex body in \(\mathbb{R}^{d+1}\) has, at least, \(d+1\) double normals.
In the paper under review, the authors consider a typical convex body (in the sense of Baire categories) \(K\subset\mathbb{R}^{d+1}\) and study the size of the set of its double normals. For instance, they prove that for most convex bodies \(K\subset\mathbb{R}^{d+1}\), the set of feet of the double normals is a Cantor set, with lower box-counting dimension \(0\) and packing dimension \(d\).
On the other hand, Kuiper also showed that set \(\mathcal{L}(K)\) of lengths of double normals of \(K\) has measure zero for \(d\leq 2\), whereas if \(d\geq 3\), there exist sets for which \(\mathcal{L}(K)\) has positive measure. Here, the authors prove that \(\mathcal{L}(K)\) is also a Cantor set for most convex bodies, having packing dimension \(1/2\) if \(d=1\), \(\geq 3/4\) if \(d=2\) and \(1\) for \(d\geq 3\).
They also get results regarding the upper and lower curvatures at the feet of the double normals of \(K\).

MSC:

52A20 Convex sets in \(n\) dimensions (including convex hypersurfaces)
54E52 Baire category, Baire spaces
28A78 Hausdorff and packing measures
28A80 Fractals

Citations:

Zbl 0131.37702
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References:

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