# Thin triangles vs Slim triangles in hyperbolic spaces

What is the difference between thin triangles and slim triangles in $\delta$ hyperbolic spaces? Google search seems to consider thin and slim as synonyms and shows the same results for the two.

• They are slightly different concepts, but turn out to encode the same notions of Gromov hyperbolicity, in the sense that: every geodesic triangle is $\delta$-thin iff every geodesic triangle is $\varepsilon$-slim, possibly with different constants. A great reference is Bridson, Haefliger, Metric spaces of non-positive curvature; have a look at def 1.1, 1.16 and Prop 1.17 in Chapter III.H – Lor Nov 19 '14 at 20:49