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<p>This paper investigates the concept of <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(L_p\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>L</mi> <mi>p</mi> </msub> </math></EquationSource> </InlineEquation>-John ellipsoids within the framework of log-concave functions. Leveraging results from the <InlineEquation ID="IEq5"> <EquationSource Format="TEX">\(L_p\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>L</mi> <mi>p</mi> </msub> </math></EquationSource> </InlineEquation> Brunn-Minkowski theory for log-concave functions, we provide a complete characterization of the <InlineEquation ID="IEq6"> <EquationSource Format="TEX">\(L_p\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>L</mi> <mi>p</mi> </msub> </math></EquationSource> </InlineEquation>-John ellipsoid and derive its associated inequalities. Furthermore, we establish an <InlineEquation ID="IEq7"> <EquationSource Format="TEX">\(L_p\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>L</mi> <mi>p</mi> </msub> </math></EquationSource> </InlineEquation> volume ratio inequality for log-concave functions, which is analogous to the classical <InlineEquation ID="IEq8"> <EquationSource Format="TEX">\(L_p\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>L</mi> <mi>p</mi> </msub> </math></EquationSource> </InlineEquation> Ball volume ratio inequality. These results extend concepts from classical convex geometry to the functional analytic setting.</p>

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\(L_p\) John Ellipsoids for Log-Concave Functions

  • 期刊论文

  • 发表日期：

    2025年12月5日

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  • Congli Yang,
  • Jianbo Fang,
  • Miao Luo,
  • Fangwei Chen

摘要

This paper investigates the concept of \(L_p\) L p -John ellipsoids within the framework of log-concave functions. Leveraging results from the \(L_p\) L p Brunn-Minkowski theory for log-concave functions, we provide a complete characterization of the \(L_p\) L p -John ellipsoid and derive its associated inequalities. Furthermore, we establish an \(L_p\) L p volume ratio inequality for log-concave functions, which is analogous to the classical \(L_p\) L p Ball volume ratio inequality. These results extend concepts from classical convex geometry to the functional analytic setting.