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

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.