New Moduli for Banach Spaces

TL;DR

This paper introduces new geometric moduli for Banach spaces based on right-angled triangles defined via quasi-orthogonality, aiming to quantify local deviations of the unit sphere from supporting hyperplanes.

Contribution

It develops novel moduli for Banach spaces using quasi-orthogonality and right-angled triangles, extending classical convexity and smoothness measures.

Findings

Proves Day-Nordlander type theorems for the new moduli

Generalizes the modulus of convexity and Banás

Provides tools for analyzing local geometry of Banach spaces

Abstract

Modifying the moduli of supporting convexity and supporting smoothness, we introduce new moduli for Banach spaces which occur, e.g., as lengths of catheti of right-angled triangles (defined via so-called quasi-orthogonality). These triangles have two boundary points of the unit ball of a Banach space as endpoints of their hypotenuse, and their third vertex lies in a supporting hyperplane of one of the two other vertices. Among other things it is our goal to quantify via such triangles the local deviation of the unit sphere from its supporting hyperplanes. We prove respective Day-Nordlander type results, involving generalizations of the modulus of convexity and the modulus of Bana\'{s}.