Higher-Order Lp Isoperimetric and Sobolev Inequalities
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866915001966526464 |
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| author | Haddad, Julián Langharst, Dylan Putterman, Eli Roysdon, Michael Ye, Deping |
| author_facet | Haddad, Julián Langharst, Dylan Putterman, Eli Roysdon, Michael Ye, Deping |
| contents | Schneider introduced an inter-dimensional difference body operator on convex bodies and proved an associated inequality. In the prequel to this work, we showed that this concept can be extended to a rich class of operators from convex geometry and proved the associated isoperimetric inequalities. The role of cosine-like operators, which generate convex bodies in $\mathbb R^n$ from those in $\mathbb R^n$, were replaced by inter-dimensional simplicial operators, which generate convex bodies in $\mathbb R^{nm}$ from those in $\mathbb R^{n}$ (or vice versa). In this work, we treat the $L^p$ extensions of these operators, and, furthermore, extend the role of the simplex to arbitrary $m$-dimensional convex bodies containing the origin. We establish $m$th-order $L^p$ isoperimetric inequalities, including the $m$th-order versions of the $L^p$ Petty projection inequality, $L^p$ Busemann-Petty centroid inequality, $L^p$ Santaló inequalities, and $L^p$ affine Sobolev inequalities. As an application, we obtain isoperimetric inequalities for the volume of the operator norm of linear functionals $(\mathbb R^n, \|\cdot\|_E) \to (\mathbb R^m, \|\cdot\|_F)$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2305_17468 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Higher-Order Lp Isoperimetric and Sobolev Inequalities Haddad, Julián Langharst, Dylan Putterman, Eli Roysdon, Michael Ye, Deping Metric Geometry Differential Geometry Functional Analysis 52A39, 52A40, 28A75 Schneider introduced an inter-dimensional difference body operator on convex bodies and proved an associated inequality. In the prequel to this work, we showed that this concept can be extended to a rich class of operators from convex geometry and proved the associated isoperimetric inequalities. The role of cosine-like operators, which generate convex bodies in $\mathbb R^n$ from those in $\mathbb R^n$, were replaced by inter-dimensional simplicial operators, which generate convex bodies in $\mathbb R^{nm}$ from those in $\mathbb R^{n}$ (or vice versa). In this work, we treat the $L^p$ extensions of these operators, and, furthermore, extend the role of the simplex to arbitrary $m$-dimensional convex bodies containing the origin. We establish $m$th-order $L^p$ isoperimetric inequalities, including the $m$th-order versions of the $L^p$ Petty projection inequality, $L^p$ Busemann-Petty centroid inequality, $L^p$ Santaló inequalities, and $L^p$ affine Sobolev inequalities. As an application, we obtain isoperimetric inequalities for the volume of the operator norm of linear functionals $(\mathbb R^n, \|\cdot\|_E) \to (\mathbb R^m, \|\cdot\|_F)$. |
| title | Higher-Order Lp Isoperimetric and Sobolev Inequalities |
| topic | Metric Geometry Differential Geometry Functional Analysis 52A39, 52A40, 28A75 |
| url | https://arxiv.org/abs/2305.17468 |