By Bart de Bruyn

Devoted to the Russian mathematician Albert Shiryaev on his seventieth birthday, this can be a number of papers written through his former scholars, co-authors and associates. The ebook represents the state of the art of a speedy maturing conception and may be a vital resource for researchers during this quarter. the variety of subject matters and accomplished form of the papers make the publication beautiful for Ph.D. scholars and younger researchers.

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**Example text**

For every i ∈ {1, 2}, let Si denote a dense slim near 2d-gon and let Fi denote a big convex subpolygon of Si . Then Ω(S1 , F1 ) ∼ = Ω(S2 , F2 ) if and only if there exists an automorphism from S1 to S2 mapping F1 to F2 . Obviously, if there exists an automorphism from S1 to S2 mapping F1 to F2 , then Ω(S1 , F1 ) ∼ = Ω(S2 , F2 ). We will prove the other direction in the following section. 40 We will take the proof from [34]. Let θ denote an isomorphism from Ω(S1 , F1 ) to Ω(S2 , F2 ). Let di (·, ·) denote the distance in Si , i ∈ {1, 2}.

If there exists a point in Γ1 (a) ∩ Γ1 (c) at distance i from x, then the lemma holds. So, we may suppose that any common neighbour of a and c is contained in Γi−1 (x). Let d1 denote a point of ab diﬀerent from a and b. Then d(d1 , c) = 2. Let d2 denote any common neighbour of d1 and c diﬀerent from b. Then d(a, d2 ) = 2. By our assumption, any point of Γ1 (c) ∩ Q at distance i − 1 from x is collinear with a. As a consequence, d2 lies at distance i from x. The path (a, d1 , d2 , c) now satisﬁes all required conditions.

Az |}, let Hi denote the unique convex subpolygon through z and the unique point of Li at distance d − 1 from z. If there would be a line zz , z ∈ Γd (x), through z not contained in any of these subpolygons, then |Az | = |Az | + |Ez,z | > |Az |, a 2 contradiction. Hence t + 1 ≤ 3d 2 (M + 1) . 38. 36. 39. Let H be a convex sub-2δ-gon of S. If there exists a (2n + 1)-cycle in Γδ (x) ∩ H for some point x ∈ H, then t + 1 ≤ (2n + 1)(M + 1) − 2n(d − δ). Proof. 29, there exists a point y at distance d(y, x) = d(y, H) = d−δ from x such that y is classical with respect to H.