Double normals of most convex bodies

TL;DR

This paper studies the typical geometric properties of convex bodies in Euclidean space, focusing on the structure and dimensions of double normals and their lengths, revealing complex fractal-like sets and curvature behaviors.

Contribution

It characterizes the fractal dimensions of the set of double normals and their lengths for typical convex bodies, and analyzes curvature properties at these double normals.

Findings

The set of feet of double normals forms a Cantor set with specific fractal dimensions.

The set of lengths of double normals is also a Cantor set with dimension zero.

The dimensions of the length set vary with the dimension d, reaching up to 1 for d≥3.

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.