Hg toolkit local geodesics and hausdorff control

Use the metric, segment and point-to-set conventions of Hg toolkit slim triangles products and four point constants. A path parametrized by arc length is a continuous map $q:I\to X$ on a real interval such that each compact restriction $q{\mathrm{\mid }}_{[s,t]}$ has length $t-s$. Here length is the supremum, over finite subdivisions $s=t_0<\cdots <t_n=t$, of ${\sum }_{i}d(q(t_{i-1}),q(t_i))$. In particular $d(q(s),q(t))\le t-s$.

For $k\ge 0$, an arc-length path is $k$-local geodesic if each restriction to a compact subinterval of length at most $k$ is an isometric parametrization. The $k=0$ condition is vacuous beyond the arc-length requirement. Results converting locality into global estimates must specify a positive radius when needed.

As in Quasi-geodesics and quasi-geodesic metric spaces, a $(\lambda ,\epsilon )$-quasi-geodesic is a map $q:I\to X$, where $\lambda \ge 1$, $\epsilon \ge 0$, satisfying ${\lambda }^{-1}\mathrm{\mid }s-t\mathrm{\mid }-\epsilon \le d(q(s),q(t))\le \lambda \mathrm{\mid }s-t\mathrm{\mid }+\epsilon$ for all $s,t\in I$. This does not assume continuity, arc-length parametrization or properness. A compact-interval segment has nonempty domain $[a,b]$ with $a\le b$; a one-point domain is allowed.

For nonempty subsets $A,B\subseteq X$, their Hausdorff distance is $d_H(A,B)=\max \{{\sup }_{a\in A}d(a,B),{\sup }_{b\in B}d(b,A)\}\mathrm{.}$ A supremum is $+\mathrm{\infty }$ if its set of values is unbounded; otherwise it exists by completeness. Thus $d_H$ takes values in $[0,+\mathrm{\infty }]$, and no finite-valued metric on all subsets is claimed. In particular $d_H(A,B)\le R$ means both point-to-set suprema are at most $R$, even if closest points do not exist. The formula excludes empty subsets. These are definitions and require no simultaneous choices.