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<p>We extend several geometric inequalities for sections and projections of convex bodies to the setting of integrable log-concave functions. In particular, we introduce suitable generalizations of the affine and dual affine quermassintegrals of a log-concave function <i>f</i> and obtain upper and lower bounds for them in terms of the integral <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\Vert f\Vert _1\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mrow> <mo stretchy="false">‖</mo> <mi>f</mi> <mo stretchy="false">‖</mo> </mrow> <mn>1</mn> </msub> </math></EquationSource> </InlineEquation>. We also establish estimates for sections and projections of log-concave functions in the spirit of the lower dimensional Busemann-Petty and Shephard problems and extend to the log-concave setting the affirmative answer to a variant of these problems, proposed by V.&#xa0;Milman. The main objective of this work is to show that the assumption of log-concavity leads to inequalities whose constants are of the same order as those in the corresponding geometric inequalities.</p>

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Inequalities for Sections and Projections of Log-Concave Functions

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  • 发表日期：

    2026年3月14日

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  • Natalia Tziotziou

摘要

We extend several geometric inequalities for sections and projections of convex bodies to the setting of integrable log-concave functions. In particular, we introduce suitable generalizations of the affine and dual affine quermassintegrals of a log-concave function f and obtain upper and lower bounds for them in terms of the integral \(\Vert f\Vert _1\) ‖ f ‖ 1 . We also establish estimates for sections and projections of log-concave functions in the spirit of the lower dimensional Busemann-Petty and Shephard problems and extend to the log-concave setting the affirmative answer to a variant of these problems, proposed by V. Milman. The main objective of this work is to show that the assumption of log-concavity leads to inequalities whose constants are of the same order as those in the corresponding geometric inequalities.