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<p>Given a finite covering by closed convex sets of <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13398_2025_1786_Article_IEq1.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="26" /> </InlineMediaObject> <EquationSource Format="TEX">\(B_X\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>B</mi> <mi>X</mi> </msub> </math></EquationSource> </InlineEquation>, the unit ball of an infinite-dimensional Banach space, we investigate whether there is a set of the covering that contains balls of radius close to 1 and (a) arbitrarily high finite dimension or (b) infinite dimension. In case (a) the answer is affirmative, but for the case (b) we just get radius close to 1/2 and finite codimension under much more restrictive hypotheses.</p>

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Covering \(B_X\) by finitely many convex sets

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  • 发表日期：

    2025年9月12日

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  • M. Raja

摘要

Given a finite covering by closed convex sets of \(B_X\) B X , the unit ball of an infinite-dimensional Banach space, we investigate whether there is a set of the covering that contains balls of radius close to 1 and (a) arbitrarily high finite dimension or (b) infinite dimension. In case (a) the answer is affirmative, but for the case (b) we just get radius close to 1/2 and finite codimension under much more restrictive hypotheses.