Metrics from valuations on convex bodies

Abstract

We associate to each valuation $\varphi$ on ${\mathcal{K}}^n$ a pseudometric $D_{\varphi }$, generalizing the classical symmetric difference and mean width metrics. Broad criteria are given under which $D_{\varphi }$ is a bona fide metric, in which case it is automatically intrinsic. Moreover, a necessary and sufficient condition for the $D_{\varphi }$-balls to be bounded in the Hausdorff sense is given; as a corollary, we obtain a condition for $D_{\varphi }$ to be proper, and hence complete and geodesic. The topology induced by $D_{\varphi }$ 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 $\mathcal{S}({\mathbb{CP}}^n,{\omega }_{FS})$ corresponding to toric model potentials, a geodesic metric that is bilipschitz equivalent to the $d_{\mathcal{S}}$ metric of Darvas, Di Nezza, and Lu.

arXiv comment53 pages, 17 figures, comments welcome