Quasisymmetric Mappings in Semimetric Spaces
An extension of quasisymmetric mappings (the Beurling-Ahlfors / Tukia-Vaisala class controlling how ratios of distances are distorted) from metric to general semimetric spaces with triangle functions. It establishes when such mappings preserve the triangle function, Ptolemy's inequality, and betweenness, gives a new diameter-ratio estimate for images of bounded sets that generalizes the Tukia-Vaisala inequality, and connects quasisymmetric mappings to weak similarities.
2501.00393
This paper generalizes quasisymmetric mappings, introduced by Beurling and Ahlfors on the real line and extended to metric spaces by Tukia and Vaisala, to the setting of general semimetric spaces (di…