Hg toolkit slim triangles products and four point constants

Work in a metric space $(X,d)$ over the real complete ordered field of Complete ordered field (least-upper-bound property). Geodesic segments and geodesic spaces have the meaning in Geodesics and geodesic metric spaces. A geodesic triangle consists of three specified segments joining three points; repeated vertices and zero-length sides are permitted. A Cayley graph here means its unit-edge geometric realization when geodesics are discussed, rather than only its vertex word metric.

For nonempty $A\subseteq X$, put $d(x,A)=\inf \{d(x,a):a\in A\}$. This infimum exists: the set is nonempty and bounded below by zero, and the infimum property follows by negating the least-upper-bound property. The closed $r$-neighbourhood means $\{x:d(x,A)\le r\}$, for $r\ge 0$; it does not assert that a closest point exists.

For $\delta \ge 0$, a triangle is $\delta$-slim if each side is contained in the closed $\delta$-neighbourhood of the union of the other two sides. A space has $\delta$-slim triangles if this holds for every choice of triangle and sides.

The Gromov product is $(x\mathrm{\mid }y{)}_o=\frac{d(o,x)+d(o,y)-d(x,y)}{2}\mathrm{.}$ The triangle inequality and its reverse give $0\le (x\mathrm{\mid }y{)}_o\le \min \{d(o,x),d(o,y)\}$. For $\kappa \ge 0$, the product condition with constant $\kappa$ is $(x\mathrm{\mid }z{)}_o\ge \min \{(x\mathrm{\mid }y{)}_o,(y\mathrm{\mid }z{)}_o\}-\kappa$ for every $o,x,y,z\in X$.

For four points $a,b,c,d$, form $d(a,b)+d(c,d)$, $d(a,c)+d(b,d)$ and $d(a,d)+d(b,c)$. The four-point condition with constant $\kappa$ says the largest minus the second-largest of these three numbers is at most $2\kappa$, for every quadruple, including repeated points. Ties are allowed. None of these definitions assumes properness or AC.