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  • $\begingroup$ What definition of "adjacent" is being used in the context of this definition? $\endgroup$
    – Lee Mosher
    Commented Aug 1, 2015 at 14:26
  • $\begingroup$ For example, if "adjacent" simply means "nonempty intersection", then a pair of 2-simplices identified at a vertex forms a nonpositively curved complex which is not F-connected. $\endgroup$
    – Lee Mosher
    Commented Aug 1, 2015 at 14:29
  • $\begingroup$ It is not exactly explained in the paper what they mean by "adjacent," but I believe nonempty intersection is correct. $\endgroup$
    – guest
    Commented Aug 1, 2015 at 17:41
  • $\begingroup$ @LeeMosher, Regarding your example, does F-connectedness fail because the inclusion map into R^2 is not an isometry? (By a rotation, we can assume the two simplices are embedded in R^2.) If so, why not? It seems that allowing the isometry to be to any subset of Euclidean space is very unrestrictive. $\endgroup$
    – guest
    Commented Aug 1, 2015 at 17:46
  • $\begingroup$ $F$-connectedness of this example would fail because there is no neighborhood of the common vertex $V$ which is isometric to a subset of Euclidean space: for all sufficiently small $r>0$, the "sphere of radius $r$" around $V$ is a union of two closed intervals $I$, $J$ such that the distance function $d(x,y)=2r$ for $x \in I$, $y \in J$ is constant. This does not happen for any subsets of Euclidean space with the Euclidean metric. $\endgroup$
    – Lee Mosher
    Commented Aug 1, 2015 at 18:36