Thanks again, I got it now.

Thanks Anton, are the triangles glued only along the side $x\overline{y}$? Are there such examples which are Cartan-Hadamard manifolds, or at least geodetically complete?

@Moishe Kohan: The normal cone has interior because it includes the normals to the spheres passing through $x$. This in turn yields that the tangent cone has nonempty interior because the tangent and normal cones are dual to each other. Finiteness of the set $X$ ensures that the points in the interior of $B'$ have finite distance from the boundary. Hence the center of $B'$ can be perturbed.