Real trees, tripod triangles, slimness and minsize

A real tree is a geodesic metric space in which every two distinct points are joined by a unique topological arc. An arc means a subspace homeomorphic to $[0,1]$, with the two specified endpoints. An injective continuous parameterization by $[0,1]$ also suffices: the interval is compact by Heine-Borel by bisection: every closed bounded interval $[a,b]$ is compact, the metric image is Hausdorff by Distinct points of a metric space have disjoint balls around them, metric and topological continuity agree by Metric continuity characterisations, with countable choice for the sequential converse, and the compact-to-Hausdorff clause of A continuous image of a compact space is compact; a continuous real-valued map on a nonempty compact space attains a maximum and a minimum; and a continuous bijection from a compact space to a Hausdorff space is a homeomorphism makes the bijection onto its image a homeomorphism.

For three vertices, a chosen geodesic triangle consists of three specified geodesic segments (Geodesics and geodesic metric spaces), allowing repeated vertices and zero-length sides. A tripod triangle is the union of three legs meeting at one branch point, with each side the union of the corresponding two legs and with distances given by the resulting tree metric; legs may have length zero.

For sides $S_1,S_2,S_3$ put

$\mathrm{slim}(\mathrm{\Delta })={\max }_i{\max }_{x\in S_i}d(x,S_j\cup S_k),\mathrm{minsize}(\mathrm{\Delta })={\min }_{x_i\in S_i}{\max }_{i,j}d(x_i,x_j)\mathrm{.}$

Here $\{i,j,k\}=\{1,2,3\}$ and $d(x,A)={\inf }_{a\in A}d(x,a)$. The extrema are justified by the following local lemma, rather than assumed from the formulas. A triangle is $\delta$-slim if its slimness is at most $\delta \ge 0$; a space is hyperbolic here if some finite $\delta$ works for every chosen triangle.

For a nonempty geodesic space and $P\ge 0$ define $m_X(P)=\sup \{\mathrm{minsize}(\mathrm{\Delta }):\mathrm{perimeter}(\mathrm{\Delta })\le P\}$. Degenerate triangles at each point ensure a nonempty family; each diameter is at most the perimeter, so $0\le m_X(P)\le P$. No profile is assigned to the empty space, which is nevertheless vacuously $\delta$-slim for every $\delta \ge 0$.

For nonempty subsets define $d_H(A,B)=\max \{{\sup }_{a\in A}d(a,B),{\sup }_{b\in B}d(b,A)\}$, allowing $+\mathrm{\infty }$. This extends the finite metric convention of Metric space: $d(x,y)=0$ iff $x=y$, symmetry, and the triangle inequality; pseudometric and ultrametric only for subset distance, not for distances between points.

In the profile formula, if the triangle has vertices $a,b,c$, then $\mathrm{perimeter}(\mathrm{\Delta })$ means the sum of the three chosen side lengths, namely $d(a,b)+d(b,c)+d(c,a)$. A zero-length side contributes $0$.