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Figure 2 | Journal of Inequalities and Applications

Figure 2

From: A combinatorial lemma and its applications

Figure 2

This figure explains the idea of the proofs of Lemma 6 and Theorem 8 for \(\pmb{n=3}\) . The vertices \(\overline{v}^{i}\), \(i\in\{1,2,3\}\), are defined by (1). The simplex \(S:= \langle{v^{1},v^{2},v^{3}} \rangle \) is triangulated with thick-lined triangles (\(K_{6}(S)\) triangulation, \(m_{1}=6\)) and then with thin-lined triangles (\(K_{12}(S)\) triangulation, \(m_{2}=2\)). The path P of simplices marked with thick-dots or thick-line is determined by adequate labeling (dependent on the theorem being considered) and combinatorial Lemma 2. For Lemma 6: there is a simplex \(\sigma\in P\) below the line \(p_{3}\geq1-\varepsilon_{1}/2\) and points \(v, v'\in\sigma\) such that \(h_{3}(v)< 0\) and \(h_{3}(v')\geq0\) (see Part 4 of the proof). For Theorem 8: the simplices in P marked with thick-line represent the compact connected set \(E_{A}\) corresponding to the given accuracy level ε and joining level-lines \(p_{3}=1/3\) and \(p_{3}=2/3\).

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