Double normals of most convex bodies

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.