Some topological properties of the intrinsic volume metric

Abstract

The purpose of this note is to derive certain basic, but previously unrecorded, topological properties of the intrinsic volume metrics $δ_1,\ldots,δ_d$ on the space of convex bodies in $\mathbb{R}^d$. Our main results show that for every $2\leq j\leq d-1$, the topology induced by $δ_j$ does not control the Hausdorff metric on the class of $j$-dimensional convex bodies; in particular, the condition $δ_j(K_n,K)\to 0$ does not imply uniform boundedness in the ambient space. Furthermore, for every $2\leq j\leq d-1$, the metric space $(K_j^d,δ_j)$ is incomplete, and remains incomplete even after adjoining the empty set. Our main results demonstrate that the intrinsic volume metric behaves in a fundamentally different way from the familiar Hausdorff and symmetric difference metrics. We describe the geometric mechanism that produces these phenomena and discuss implications for geometric tomography, metric stability theory and integral geometry.

Theorems & Definitions (7)

• Theorem 1.1: Failure of uniform boundedness
• Theorem 1.2: Incompleteness
• Theorem 1.3: Incompleteness with $\varnothing$ adjoined
• Remark 1.4
• Remark 2.1
• Lemma 3.1
• proof : Proof of Lemma \ref{['lem:a.eHgood']}