<p>In this paper, two new geometric constants <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$ BS(X) $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>B</mi> <mi>S</mi> <mo stretchy="false">(</mo> <mi>X</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$ SI(X) $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>S</mi> <mi>I</mi> <mo stretchy="false">(</mo> <mi>X</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> are introduced to quantify the differences between Birkhoff, Singer, and isosceles orthogonalities in Banach spaces. Our main results establish quantitative relationships between these constants and key geometric properties. Specifically, we prove that <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$ BS(X)<2 $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>B</mi> <mi>S</mi> <mo stretchy="false">(</mo> <mi>X</mi> <mo stretchy="false">)</mo> <mo><</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation> implies uniform normal structure, and that <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$ SI(X)<\sqrt{2} $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>S</mi> <mi>I</mi> <mrow> <mo stretchy="false">(</mo> <mi>X</mi> <mo stretchy="false">)</mo> </mrow> <mo><</mo> <msqrt> <mn>2</mn> </msqrt> </mrow> </math></EquationSource> </InlineEquation> implies uniform non-squareness. Furthermore, we show that both uniformly convex and uniformly smooth spaces satisfy these inequalities. This study thus provides a new perspective into the geometry of Banach spaces through the study of orthogonality relations.</p>

Two Geometric Constants Measuring Differences Between Orthogonality Types in Banach Spaces

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发表日期：

2025年12月8日

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Jiaye Bi,
Qi Liu,
Yongjin Li

摘要

In this paper, two new geometric constants $ BS(X) $ $B S (X)$ and $ SI(X) $ $S I (X)$ are introduced to quantify the differences between Birkhoff, Singer, and isosceles orthogonalities in Banach spaces. Our main results establish quantitative relationships between these constants and key geometric properties. Specifically, we prove that $ BS(X)<2 $ $B S (X) < 2$ implies uniform normal structure, and that $ SI(X)<\sqrt{2} $ $S I (X) < \sqrt{2}$ implies uniform non-squareness. Furthermore, we show that both uniformly convex and uniformly smooth spaces satisfy these inequalities. This study thus provides a new perspective into the geometry of Banach spaces through the study of orthogonality relations.