A non-geodesic analogue of Reshetnyak's majorization theorem (Q6113011)

Given \(\kappa \in \mathbb{R}\) and an integer \(n \ge 4\), the following condition was introduced by \textit{M. Gromov} [J. Math. Sci., New York 119, No. 2, 178--200 (2001; Zbl 1089.53029); translation from Zap. Nauchn. Semin. POMI 280, 101--140 (2001)] in connection to the study of metric spaces that can be isometrically embedded into a \(\mathrm{CAT}(\kappa)\)-space. Definition. Fix \(\kappa\in\mathbb{R}\) and an integer \(n \ge 4\). Let \((X,d_X )\) be a metric space. We say that \(X\) satisfies the \(\mathrm{Cycl}_n (\kappa )\) condition if for any map \(f:\mathbb Z / n\mathbb Z\to X\) with \[ \sum_{i\in\mathbb{Z}/n\mathbb{Z}}d_X \left(f(i),f(i+\lbrack 1\rbrack_n )\right) <2 D_{\kappa}, \] there exists a map \(g:\mathbb Z / n\mathbb Z\to M_{\kappa}^2\) such that \[ d_{\kappa}(g(i),g(i+\lbrack 1\rbrack_n ))\leq d_X (f(i),f(i+\lbrack 1\rbrack_n )),\quad d_{\kappa}(g(i),g(j))\geq d_X (f(i),f(j)) \] for any \(i,j\in\mathbb{Z}/n\mathbb{Z}\) with \(j\neq i+\lbrack 1\rbrack_n\) and \(i\neq j+\lbrack1\rbrack_n\). (The notation \(i+\lbrack 1\rbrack_n\) used in the paper is meant to be \(1\) if \(i=n\) and \(i+1\) otherwise.) Note that \(d_{\kappa}\) stands for the distance function on \(M_{\kappa}^2\), the complete, simply connected, two-dimensional Riemannian manifold of constant sectional curvature \(\kappa\). Every \(\mathrm{CAT}(\kappa)\)-space satisfies the \(\mathrm{Cycl}_n (\kappa )\) condition for all integers \(n \ge 4\). The \(\mathrm{Cycl}_4 (\kappa )\) condition characterizes \(\mathrm{CAT}(\kappa)\)-spaces among geodesic metric spaces, but, in the context of a metric space that is not geodesic, it cannot guarantee alone its isometric embeddability into a \(\mathrm{CAT}(\kappa)\)-space. The main result can be interpreted as a variant of \textit{Yu. G. Reshetnyak}'s majorization theorem [Sib. Math. J. 9, 683--689 (1968; Zbl 0176.19503)] in the setting of metric spaces with the \(\mathrm{Cycl}_4 (\kappa )\) condition. Theorem. Let \(\kappa\in\mathbb{R}\). If a metric space \((X,d_X)\) satisfies the \(\mathrm{Cycl}_4 (\kappa )\) condition, then for any integer \(n\geq 3\), and for any map \(f:\mathbb{Z}/n\mathbb{Z}\to X\) with \[ \sum_{i\in\mathbb{Z}/n\mathbb{Z}}d_X \left( f(i),f(i+\lbrack 1\rbrack_n )\right)<2 D_{\kappa},\quad f(j)\neq f(j+\lbrack 1\rbrack_n ) \] for every \(j\in\mathbb{Z}/n\mathbb{Z}\), there exists a map \(g:\mathbb{Z}/n\mathbb{Z}\to M_{\kappa}^2\) such that the following hold: \begin{itemize} \item[(1)] For any \(i,j\in\mathbb{Z}/n\mathbb{Z}\), we have \[ d_{\kappa}(g(i),g(i+\lbrack 1\rbrack_n )) =d_X (f(i),f(i+\lbrack 1\rbrack_n )),\quad d_{\kappa}(g(i),g(j))\geq d_X (f(i),f(j)). \] \item[(2)] For any \(i,j\in\mathbb{Z}/n\mathbb{Z}\) with \(i\neq j\), we have \(\lbrack g(i),g(j)\rbrack\cap\lbrack g(i-\lbrack 1\rbrack_n ),g(i+\lbrack 1\rbrack_n )\rbrack\neq\emptyset\), where \(\lbrack a,b\rbrack\) is the line segment in \(M_{\kappa}^2\) with endpoints \(a\) and \(b\). \end{itemize} This allows the author to deduce that, in any metric space, the \(\mathrm{Cycl}_4 (\kappa )\) condition yields the \(\mathrm{Cycl}_n (\kappa )\) condition for all integers \(n \ge 4\). The paper also discusses some consequences assuming that the weighted quadruple inequalities hold.