Farthest points on flat surfaces

Rouyer, Joël; Vîlcu, Costin

doi:10.1007/s00022-018-0448-9

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“…Our work is also related to the articles [3], [29], [30], [31], which focus on intersections of infinitely many affine diameters or normals for typical convex bodies. Let us mention here that, for d ≥ 2, double normals of a typical convex body are pairwise disjoint [24]. For other Baire categories results about convex bodies, see e.g.…”

Section: Introduction and Resultsmentioning

confidence: 99%

Double normals of most convex bodies

Rivière

Rouyer

Vîlcu

et al. 2019

Advances in Mathematics

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We consider a typical (in the sense of Baire categories) convex body K in 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 1 2 if d = 1, is at least 3 4 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.MSC 2010: 52A20, 54E52, 28A78, 28A80

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