Some inequalities related to Heinz mean constant with Birkhoff orthogonality

TL;DR

This paper introduces a new geometric constant in Banach spaces based on Heinz means and Birkhoff orthogonality, providing bounds, characterizations, and relations to known geometric properties.

Contribution

It defines a novel Heinz mean constant for Banach spaces, explores its bounds, and characterizes specific geometric structures like Radon planes and uniformly non-square spaces.

Findings

Derived bounds for the Heinz mean constant in Banach spaces.

Characterized the constant's value in Hilbert spaces.

Linked the constant to properties like uniform non-squareness and Radon planes.

Abstract

Motivated by the work of Baronti et al. [J. Math. Anal. Appl. 252(2000) 124-146], where they defined the supremum of an arithmetic mean of the side lengths of a triangle, summing antipodal points on the unit sphere, we introduce a new geometric constant for Banach spaces, utilizing the Heinz means that interpolate between the geometric and arithmetic means associated with Birkhoff orthogonality. We discuss the bounds in Banach spaces and find the values of constant in Hilbert spaces. We obtain the characterization of uniformly non-square spaces. We investigate the correlation between our notion of the Heinz mean constant and other well-known terms, viz., the modulus of convexity, modulus of smoothness, and rectangular constant. Furthermore, we also give a characterization of the Radon plane with an affine regular hexagonal unit sphere.