Double normals of most convex bodies

We consider a typical (in the sense of Baire categories) convex body K in Rd+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...

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Bibliographic Details
Published in:Advances in mathematics (New York. 1965) 2019-02, Vol.343, p.245-272
Main Authors: Rivière, Alain, Rouyer, Joël, Vîlcu, Costin, Zamfirescu, Tudor
Format: Article
Language:English
Subjects:
Baire categories
Box dimension
Classical Analysis and ODEs
Double normals of a convex body
Mathematics
Packing dimension
Upper and lower curvature
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Summary:We consider a typical (in the sense of Baire categories) convex body K in Rd+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 12 if d=1, is at least 34 if d=2, and equals 1 if d≥3. 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.
ISSN:0001-8708
1090-2082
DOI:10.1016/j.aim.2018.11.014