Geometric Characterisation of Generalised n-Normed Spaces: Uniform n-Convexity, n-Smoothness and Reflexivity

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@article{1723226,
      author = {Anyande Benard Alex, Shem Aywa, Patrick Makila Wanjala},
      title = {Geometric Characterisation of Generalised n-Normed Spaces: Uniform n-Convexity, n-Smoothness and Reflexivity},
      journal = {Iconic Research And Engineering Journals},
      year = {2026},
      volume = {10},
      number = {3},
      pages = {2420-2423},
      issn = {2456-8880},
      url = {https://www.irejournals.com/formatedpaper/1723226.pdf},
      abstract = {This paper develops the geometric theory of generalised n-normed spaces with the aim of characterising those geometric properties that are sufficient for the existence of fixed points of nonexpansive mappings. We study the modulus of n-convexity δ_X^((n) ) and the modulus of n-smoothness ρ_X^((n) ), establishing their basic quantitative properties, an n-dimensional parallelogram identity for n-inner product spaces, and the n-dimensional analogues of the Clarkson inequalities. We prove that strict n-convexity yields uniqueness of best n-approximations, and we obtain an n-dimensional Lindenstrauss-type duality inequality relating δ_X^((n) ) and ρ_(X^*)^((n) ), from which the duality between uniform n-convexity and uniform n-smoothness follows. Finally, we establish an n-normed Milman–Pettis theorem—every uniformly n-convex n-Banach space is reflexive—together with the weak compactness of bounded closed convex sets and the fact that uniform n-convexity implies n-normal structure. These results identify uniform n-convexity as the central geometric hypothesis underlying the fixed point theory of nonexpansive mappings in the n-normed setting.},
      keywords = {Modulus of n-convexity, n-Normed space, Reflexivity, Strict n-convexity, Uniform n-convexity.},
      month = {September},
      doi = {https://doi.org/10.64388/IREV10I3-1723226}
  }