Metrics from valuations on convex bodies

Abstract

We associate to each valuation $ϕ$ on $\mathcal{K}^n$ a pseudometric $D_ϕ$, generalizing the classical symmetric difference and mean width metrics. Broad criteria are given under which $D_ϕ$ is a bona fide metric, in which case it is automatically intrinsic. Moreover, a necessary and sufficient condition for the $D_ϕ$-balls to be bounded in the Hausdorff sense is given; as a corollary, we obtain a condition for $D_ϕ$ to be proper, and hence complete and geodesic. The topology induced by $D_ϕ$ is also investigated: we provide conditions relating different forms of convergence, showing for instance that the intrinsic volumes $\left \{\text{V}_j : j \in \{1, \dots, n\} \right\}$ induce metrics that are topologically equivalent to the Hausdorff distance. Finally, we apply our construction to the metric geometry of singularity types by exhibiting, on the subspace of $\mathcal{S}(\mathbb{CP}^n,ω_{FS})$ corresponding to toric model potentials, a geodesic metric that is bilipschitz equivalent to the $d_{\mathcal{S}}$ metric of Darvas, Di Nezza, and Lu.