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<p>In this paper, we deal with generalizations of the Mahler volume product for log-concave functions. We show that the polarity transform <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11784_2025_1221_Article_IEq1.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="18" /> </InlineMediaObject> <EquationSource Format="TEX">\({\mathcal {A}}\)</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">A</mi> </math></EquationSource> </InlineEquation> can be rescaled so that the Mahler product it induces has upper and lower bounds of the same asymptotics. We discuss a similar result for the <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11784_2025_1221_Article_IEq2.gif" Format="GIF" Height="15" Rendition="HTML" Resolution="72" Type="Linedraw" Width="17" /> </InlineMediaObject> <EquationSource Format="TEX">\({\mathcal {J}}\)</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">J</mi> </math></EquationSource> </InlineEquation> transform. As an application, we extend the König–Milman duality of entropy result to the class of geometric log-concave functions.</p>

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The scaled polarity transform and related inequalities

  • 期刊论文

  • 发表日期：

    2025年7月28日

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  • Shoni Gilboa,
  • Alexander Segal,
  • Boaz A. Slomka

摘要

In this paper, we deal with generalizations of the Mahler volume product for log-concave functions. We show that the polarity transform \({\mathcal {A}}\) A can be rescaled so that the Mahler product it induces has upper and lower bounds of the same asymptotics. We discuss a similar result for the \({\mathcal {J}}\) J transform. As an application, we extend the König–Milman duality of entropy result to the class of geometric log-concave functions.