The weak min-max property in Banach spaces


  • Zhengyong Ouyang
  • Antti Rasila
  • Tiantian Guan



In this paper, we investigate the relationship between the weak min-max property and the diameter uniformity of domains in Banach spaces with dimension at least 2. As an application, we show that diameter uniform domains are invariant under relatively quasimöbius mappings.


Gehring, F. W., and Hag, K., Remarks on uniform and quasiconformal extension domains, Complex Variables Theory Appl. 9 (1987), no. 2–3, 175–188.

Gehring, F. W., and Osgood, B. G., Uniform domains and the quasi-hyperbolic metric, J. Anal. Math. 36 (1979), 50–74.

Gehring, F. W., and Palka, B. P., Quasiconformally homogeneous domains, J. Anal. Math. 30 (1976), 172–199.

Huang, M., Li, Y., Vuorinen, M., and Wang, X., On quasimöbius maps in real Banach spaces, Israel J. Math. 198 (2013), no. 1, 467–486.

Jones, P. W., Extension theorems for BMO, Indiana Univ. Math. J. 29 (1980), no. 1, 41–66.

Li, Y., Ponnusamy, S., and Zhou, Q., Sphericalization and flattening preserve uniform domains in non-locally compact metric spaces, J. Aust. Math. Soc. 112 (2022), no. 1, 68–89.

Li, Y., Vuorinen, M., and Zhou, Q., Weakly quasisymmetric maps and uniform spaces, Comput. Methods Funct. Theory 18 (2018), no. 4, 689–715.

Li, Y., Vuorinen, M., and Zhou, Q., Apollonian metric, uniformity and Gromov hyperbolicity, Complex Var. Elliptic Equ. 65 (2020), no. 2, 215–228.

Martio, O., Definitions of uniform domains, Ann. Acad. Sci. Fenn. Ser. A I Math. 5 (1980), 197–205.

Martio, O., and Sarvas, J., Injectivity theorems in plane and space, Ann. Acad. Sci. Fenn. Ser. A I Math. 4 (1978), no. 1, 383–401.

Väisälä, J., Quasi-Möbius maps, J. Anal. Math. 44 (1984/85), 218–234.

Väisälä, J., Uniform domains, Tohoku Math. J. (2) 40 (1988), no. 1, 101–118.

Väisälä, J., Free quasiconformality in Banach spaces. II, Ann. Acad. Sci. Fenn. Ser. A I Math. 16 (1991), no. 2, 255-310.

Väisälä, J., The free quasiworld. Freely quasiconformal and related maps in Banach spaces, Quasiconformal geometry and dynamics (Lublin 1996), 55–118, Banach Center Publ. 48, Polish Acad. Sci. Inst. Math., Warsaw, 1999.

Väisälä, J., Broken tubes in Hilbert spaces, Analysis (Munich) 24 (2004), no. 3, 227–238.

Vuorinen, M., Capacity densities and angular limits of quasiregular mappings, Trans. Amer. Math. Soc. 263 (1981), no. 2, 343–354.

Zhou, Q., and Rasila, A., Quasimöbius invariance of uniform domains, Studia Math. 261 (2021), no. 1, 1–24.



How to Cite

Ouyang, Z., Rasila, A., & Guan, T. (2022). The weak min-max property in Banach spaces: Array. MATHEMATICA SCANDINAVICA, 128(2).