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By Alexander Brudnyi, Yuri Brudnyi

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23) ρ0 (I (v ), I (w)) ≤ log n · d(v , w) and it remains to establish the inverse inequality. To this end we consider two cases. First, suppose that w = a(v , w). 24) ρ0 (c(v ), c(w)) ≥ log 1 + |c2 (v ) − c2 (w)| min (c2 (v ), c2 (w)) = log nlv −lw = d(v , w) log n. 21). To estimate the second term one notes that, by our geometric construction, the orthogonal projection of Q(v ) onto the bottom side of Q(v + ) lies inside this side. 23), and taking into account that vi+1 = vi+ and wi+1 = w+i we conclude that the orthogonal projections of c(v ) := c(v1 ) and c(w) := c(w1 ) onto the bottom side of + ) = Q(w+ ) lie, respectively, inside the top sides of the Q(a(v , w)) = Q(vp−1 q−1 squares Q(vp−1 ) and Q(wq−1 ) adjoint to Q(a(v , w)).

22) λ(S, M ) := inf{ E : E ∈ Ext(S, M )}; here we set λ(S, M ) := 0, if S = ∅. 5. 23) sup λ(Fi , M ) = ∞. 24) sup λ(Fi \ B, M ) = ∞. i Proof. 1, we can assume that m∗ ∈ Fi , i ∈ N. By the same reason we may and will assume that all Fi contain a fixed point m ∈ M \ B. 25) Ei1 ∈ Ext(Fi \ B, M ) and Ei1 ≤ A1 ; here we set Ei1 := 0, if Fi \ B = ∅. Let 2B be the open ball centered at m∗ and of twice the radius of B. 26) U1 := M \ B and U2 := 2B, and let {ρ1 , ρ2 } be the corresponding Lipschitz partition of unity (cf.

12) B(v ) := {m ∈ MΓ : d(m, v ) < 3}. So B(v ) is the union of at most three intervals of length 3 each of which has the form e ∩ B(v ) where every e belongs to the set of edges E(v ) incident to v . We enumerate these intervals by numbers from the set ω ⊂ {1, 2, 3} where ω = {1}, {1, 2} or {1, 2, 3}, if deg v = 1, 2 or 3, respectively. This set of indices will be denoted by ω(v ) and i(e, v ) (briefly, i(e)) will stand for the number of e ∩ B(v ) in this enumeration. We then introduce a coordinate system ψv : B(v ) → R3 , v ∈ V, of MΓ as follows.

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