Some aspects of generalized Dunkl-Williams constant in Banach spaces
Haoyu Zhou, Qi Liu, Yuxin Wang
TL;DR
The paper introduces generalized Dunkl-Williams constants $DW(X,α,β)$ and $DW_B(X,α,β)$ in real Banach spaces and derives sharp upper and lower bounds, establishing how these constants quantify the space's deviation from Hilbert geometry. It provides two equivalent characterizations of each constant, links them to classical geometric constants such as the James constant $J(X)$, the modulus of convexity $δ_X$, its convexity index $ε_0(X)$, and the Lindenstrauss smoothness $ρ′(0)$, and studies their behavior under isomorphisms and in special spaces like $X_μ$. The paper also introduces the Birkhoff-orthogonality based constant $DW_B(X,α,β)$, gives tight bounds in terms of the rectangular constant $μ(X)$, and presents an explicit computation in $( ext{R}^2, orm{ullet}_∞)$. Together, these results deepen the understanding of angular distance measures in Banach spaces and their connections to convexity, smoothness, and Hilbert-space proximity.
Abstract
This article delves into an exploration of two innovative constants, namely DW(X,α,\b{eta}) and DWB (X,α,\b{eta}), both of which constitute extensions of the Dunkl-Williams constant. We derive both the upper and lower bounds for these two constants and establish two equivalent relations between them. Moreover, we elucidate the relationships between these constants and several well-known constants. Additionally, we have refined the value of the DWB (X,α,\b{eta}) constant in certain specific Banach spaces.