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<p>We prove the Shannon’s inequality on non-collapsing <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12220_2025_1918_Article_IEq3.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="77" /> </InlineMediaObject> <EquationSource Format="TEX">\(\textsf{RCD}(0,N)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="sans-serif">RCD</mi> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mi>N</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> spaces. In the proof, we use the characterization of the <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12220_2025_1918_Article_IEq4.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="47" /> </InlineMediaObject> <EquationSource Format="TEX">\(\textsf{EVI}_{0,N}\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="sans-serif">EVI</mi> <mrow> <mn>0</mn> <mo>,</mo> <mi>N</mi> </mrow> </msub> </math></EquationSource> </InlineEquation>-gradient flow of the relative entropy and the infinitesimal behavior of the heat kernel. Also, we have a cone rigidity result. As an application, we have the so-called uncertainty principle inequality on such spaces.</p>

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Shannon’s Inequality on Non-collapsing \(\textsf{RCD}\) Spaces

  • 期刊论文

  • 发表日期：

    2025年2月6日

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  • Yu Kitabeppu

摘要

We prove the Shannon’s inequality on non-collapsing \(\textsf{RCD}(0,N)\) RCD ( 0 , N ) spaces. In the proof, we use the characterization of the \(\textsf{EVI}_{0,N}\) EVI 0 , N -gradient flow of the relative entropy and the infinitesimal behavior of the heat kernel. Also, we have a cone rigidity result. As an application, we have the so-called uncertainty principle inequality on such spaces.