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<p>We present a direct relationship between the Shannon entropy of upper level set volumes and the Rényi entropies of non-negative, compactly supported continuous functions. Using this foundational connection, we derive anisotropic <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12220_2025_1945_Article_IEq1.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="22" /> </InlineMediaObject> <EquationSource Format="TEX">\(\log \)</EquationSource> <EquationSource Format="MATHML"><math> <mo>log</mo> </math></EquationSource> </InlineEquation>-moment-entropy inequalities and reveal their links to the dual <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12220_2025_1945_Article_IEq1.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="22" /> </InlineMediaObject> <EquationSource Format="TEX">\(\log \)</EquationSource> <EquationSource Format="MATHML"><math> <mo>log</mo> </math></EquationSource> </InlineEquation>-Minkowski inequality.</p>

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Anisotropic \(\log \)-Moment-Entropy Inequalities

  • 期刊论文

  • 发表日期：

    2025年3月4日

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  • Songjun Lv

摘要

We present a direct relationship between the Shannon entropy of upper level set volumes and the Rényi entropies of non-negative, compactly supported continuous functions. Using this foundational connection, we derive anisotropic \(\log \) log -moment-entropy inequalities and reveal their links to the dual \(\log \) log -Minkowski inequality.