Maligranda introduced the p-angular distance in 2006, and Rooin proposed the skew p-angular distance in 2018. This paper introduces the Maligranda-Rooin constant, a new geometric constant in Banach spaces to compare these two distances. We denote the Maligranda-Rooin constant as <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\mathcal{M}\mathcal{R}_p(\mathcal {X})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="script">M</mi> <msub> <mi mathvariant="script">R</mi> <mi>p</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="script">X</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>. First, the bounds of <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\mathcal{M}\mathcal{R}_p(\mathcal {X})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="script">M</mi> <msub> <mi mathvariant="script">R</mi> <mi>p</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="script">X</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> are derived. Next, it is shown that a normed linear space is an inner product space if and only if <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\mathcal{M}\mathcal{R}_p(\mathcal {X})=1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="script">M</mi> <msub> <mi mathvariant="script">R</mi> <mi>p</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="script">X</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>. Furthermore, an equivalent form of this new constant is established. Finally, the relationships between <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\mathcal{M}\mathcal{R}_p(\mathcal {X})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="script">M</mi> <msub> <mi mathvariant="script">R</mi> <mi>p</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="script">X</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> and uniform non-squareness, uniform convexity, uniform smoothness, as well as uniform normal structure are investigated.

The Constant to Measure the Differences Between p-Angular Distance and Skew p-Angular Distance

期刊论文
发表日期：

2026年7月21日

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Qi Liu,
Yuxin Wang

摘要

Maligranda introduced the p-angular distance in 2006, and Rooin proposed the skew p-angular distance in 2018. This paper introduces the Maligranda-Rooin constant, a new geometric constant in Banach spaces to compare these two distances. We denote the Maligranda-Rooin constant as $\mathcal{M}\mathcal{R}_p(\mathcal {X})$ $M R_{p} (X)$ . First, the bounds of $\mathcal{M}\mathcal{R}_p(\mathcal {X})$ $M R_{p} (X)$ are derived. Next, it is shown that a normed linear space is an inner product space if and only if $\mathcal{M}\mathcal{R}_p(\mathcal {X})=1$ $M R_{p} (X) = 1$ . Furthermore, an equivalent form of this new constant is established. Finally, the relationships between $\mathcal{M}\mathcal{R}_p(\mathcal {X})$ $M R_{p} (X)$ and uniform non-squareness, uniform convexity, uniform smoothness, as well as uniform normal structure are investigated.