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<p>The main aim of this article is to establish certain characterizations of rotund spaces, uniformly rotund spaces and (compactly) locally uniformly rotund spaces. To achieve this, we present the idea of property <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40840_2025_1922_Article_IEq1.gif" Format="GIF" Height="15" Rendition="HTML" Resolution="72" Type="Linedraw" Width="33" /> </InlineMediaObject> <EquationSource Format="TEX">\(UC'\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>U</mi> <msup> <mi>C</mi> <mo>′</mo> </msup> </mrow> </math></EquationSource> </InlineEquation> and a few of its generalizations. First, we establish various interrelationships among these concepts by utilizing remotality notions and provide several counterexamples. We demonstrate that a space is uniformly rotund if and only if every pair of non-empty, closed, bounded subsets has property <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40840_2025_1922_Article_IEq1.gif" Format="GIF" Height="15" Rendition="HTML" Resolution="72" Type="Linedraw" Width="33" /> </InlineMediaObject> <EquationSource Format="TEX">\(UC'\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>U</mi> <msup> <mi>C</mi> <mo>′</mo> </msup> </mrow> </math></EquationSource> </InlineEquation> if and only if the pair of unit spheres has property <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40840_2025_1922_Article_IEq1.gif" Format="GIF" Height="15" Rendition="HTML" Resolution="72" Type="Linedraw" Width="33" /> </InlineMediaObject> <EquationSource Format="TEX">\(UC'\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>U</mi> <msup> <mi>C</mi> <mo>′</mo> </msup> </mrow> </math></EquationSource> </InlineEquation>. Similar characterizations for various generalizations of uniform rotundity are also provided.</p>

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Characterizations of some Rotundity Properties using Property \(UC'\) and its Generalizations

  • 期刊论文

  • 发表日期：

    2025年7月14日

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  • C. Arunachala Prasath,
  • Vamsinadh Thota

摘要

The main aim of this article is to establish certain characterizations of rotund spaces, uniformly rotund spaces and (compactly) locally uniformly rotund spaces. To achieve this, we present the idea of property \(UC'\) U C ′ and a few of its generalizations. First, we establish various interrelationships among these concepts by utilizing remotality notions and provide several counterexamples. We demonstrate that a space is uniformly rotund if and only if every pair of non-empty, closed, bounded subsets has property \(UC'\) U C ′ if and only if the pair of unit spheres has property \(UC'\) U C ′ . Similar characterizations for various generalizations of uniform rotundity are also provided.