A new geometric constant to compare p-angular and skew p-angular distances
Yuxin Wang, Qi Liu, Jinyu Xia, Muhammad Sarfraz
TL;DR
The paper introduces the Maligranda-Rooin constant $\\mathcal{MR}_p(\\mathcal{X})$ to quantify how the $p$-angular distance $\\alpha_p$ compares with the skew $p$-angular distance $\\beta_p$ in real Banach spaces. It proves a sharp bound $1 \le\\mathcal{MR}_p(\\mathcal{X})\le 2$, provides an equivalent form and shows $\\mathcal{MR}_p(\\mathcal{X})=1$ iff $\\mathcal{X}$ is an inner product space, and notes the special cases $\\mathcal{MR}_1(\\mathcal{X})=1$ and $\\mathcal{MR}_0(\\mathcal{X})=\\mathcal{DR}(\\mathcal{X})$. The work connects this constant to uniform non-squareness and normal structure, giving lower bounds in terms of the modulus of convexity and smoothness and establishing normal-structure criteria via $\\mu(\\mathcal{X})$ and $\\mathcal{M}(\\mathcal{X})$. These results provide a quantitative bridge between angular-distance geometry and core Banach-space properties such as uniform convexity, smoothness, and inner-product structure.
Abstract
The $p$-angular distance was first introduced by Maligranda in 2006, while the skew $p$-angular distance was first introduced by Rooin in 2018. In this paper, we shall introduce a new geometric constant named Maligranda-Rooin constant in Banach spaces to compare $p$-angular distance and skew $p$-angular distance. We denote the Maligranda-Rooin constant as $\mathcal{M} \mathcal{R}_p(\mathcal{X})$. First, the upper and lower bounds for the $\mathcal{M} \mathcal{R}_p(\mathcal{X})$ constant is given. Next, it's shown that, a normed linear space is an inner space if and only if $\mathcal{M} \mathcal{R}_p(\mathcal{X})=1$. Moreover, an equivalent form of this new constant is established. By means of the $\mathcal{M} \mathcal{R}_p(\mathcal{X})$ constant, we carry out the quantification of the characterization of uniform nonsquareness. Finally, we study the relationship between the $\mathcal{M} \mathcal{R}_p(\mathcal{X})$ constant, uniform convexity, uniform smooth and normal structure.