Higher-Order Reverse Isoperimetric Inequalities for Log-concave Functions
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| Format: | Preprint |
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2024
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| _version_ | 1866917061930778624 |
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| author | Langharst, Dylan Sola, Francisco Marín Ulivelli, Jacopo |
| author_facet | Langharst, Dylan Sola, Francisco Marín Ulivelli, Jacopo |
| contents | The Rogers-Shephard and Zhang's projection inequalities are two reverse, affine isoperimetric-type inequalities for convex bodies. Following a classical work by Schneider, both inequalities have been extended to the so-called $m$th-order setting. In this work, we establish the $m$th-order analogues for these inequalities in the setting of log-concave functions. Our proof of the functional Zhang's projection inequality employs properties of the asymmetric LYZ body, significantly streamlining the argument and producing a novel approach for the case $m=1$. Furthermore, we introduce and analyze the radial mean bodies of a log-concave function, thereby providing a functional generalization of Gardner and Zhang's radial mean bodies. These are new even in the case $m=1$. Our development leverages an extension of Ball bodies, which may be of independent interest. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2403_05712 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Higher-Order Reverse Isoperimetric Inequalities for Log-concave Functions Langharst, Dylan Sola, Francisco Marín Ulivelli, Jacopo Metric Geometry Functional Analysis Primary: 26B25, 52A38, 52A40, Secondary: 52A20 The Rogers-Shephard and Zhang's projection inequalities are two reverse, affine isoperimetric-type inequalities for convex bodies. Following a classical work by Schneider, both inequalities have been extended to the so-called $m$th-order setting. In this work, we establish the $m$th-order analogues for these inequalities in the setting of log-concave functions. Our proof of the functional Zhang's projection inequality employs properties of the asymmetric LYZ body, significantly streamlining the argument and producing a novel approach for the case $m=1$. Furthermore, we introduce and analyze the radial mean bodies of a log-concave function, thereby providing a functional generalization of Gardner and Zhang's radial mean bodies. These are new even in the case $m=1$. Our development leverages an extension of Ball bodies, which may be of independent interest. |
| title | Higher-Order Reverse Isoperimetric Inequalities for Log-concave Functions |
| topic | Metric Geometry Functional Analysis Primary: 26B25, 52A38, 52A40, Secondary: 52A20 |
| url | https://arxiv.org/abs/2403.05712 |