Double normals of most convex bodies

2018/04/19 by Rivière, Alain, Rouyer, Joël, Vîlcu, Costin +1
#28A78 #28A80 #52A20 #54E52 #FOS: Mathematics #Metric Geometry (math.MG)

paper · doi:10.48550/arxiv.1804.07015

Abstract

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.

Related

• Typical curvature behaviour of bodies of constant width
• Magic mirrors, dense diameters, Baire category
• On a normed version of a Rogers-Shephard type problem
• Upper bound for the Dvoretzky dimension in Milman-Schechtman theorem
• The shattering dimension of sets of linear functionals