On certain optimal diffeomorphisms between closed curves (Q470226)

In \textit{size theory}, the pattern recognition side of persistent homology, shape is formalized not as a space, but as a \textit{size pair} \((X, \varphi)\) of a topological space and a map defined on it, mostly (and here in particular) with \(\mathbb R\) as a range [\textit{S. Biasotti}, \textit{L. De Floriani}, \textit{B. Falcidieno}, \textit{P. Frosini}, \textit{D. Giorgi}, \textit{C. Landi}, \textit{L. Papaleo} and \textit{M. Spagnuolo}, Describing shapes by geometrical-topological properties of real functions, ACM Computing Surveys, 40(4), 12:1-12:87 (2008)].NEWLINENEWLINEA pseudodistance between size pairs \((X, \varphi)\), \((Y, \psi)\), where \(X\) and \(Y\) are homeomorphic, is defined as follows. For any homeomorphism \(f: X \to Y\) let \(\Theta(f) = max_{x\in X} |\varphi(x)-\psi(f(x))|\). Then the \textit{natural pseudodistance} \(\delta \big( (X, \varphi), (Y, \psi)\big)\) is defined as the infimum of \(\{\Theta(f)\}\) for \(f\) in the set of all such homeomorphisms \textit{P. Frosini} and \textit{M. Mulazzani} [Bull. Belg. Math. Soc. - Simon Stevin 6, No. 3, 455--464 (1999; Zbl 0937.55010)]. \(f: X \to Y\) is said to be an \textit{optimal} homeomorphism if \(\Theta(f) = \delta \big( (X, \varphi), (Y, \psi)\big)\).NEWLINENEWLINEDoes an optimal homeomorphism always exist? Yes, and it is of class \({\mathcal C}^2\), if the two spaces are curves, the two maps are Morse functions and the pseudodistance vanishes (Theorem 3.4). No, if any of these three conditions is missing, as the authors show by smart examples.