On the $m$th order $p$-affine capacity
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866912381458710528 |
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| author | Zhou, Xia Ye, Deping |
| author_facet | Zhou, Xia Ye, Deping |
| contents | Let $M_{n, m}(\mathbb{R})$ denote the space of $n\times m$ real matrices, and $\mathcal{K}_o^{n,m}$ be the set of convex bodies in $M_{n, m}(\mathbb{R})$ containing the origin. We develop a theory for the $m$th order $p$-affine capacity $C_{p,Q}(\cdot)$ for $p\in[1,n)$ and $Q\in\mathcal{K}_{o}^{1,m}$. Several equivalent definitions for the $m$th order $p$-affine capacity will be provided, and some of its fundamental properties will be proved, including for example, translation invariance and affine invariance. We also establish several inequalities related to the $m$th order $p$-affine capacity, including those comparing to the $p$-variational capacity, the volume, the $m$th order $p$-integral affine surface area, as well as the $L_p$ surface area. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_12573 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the $m$th order $p$-affine capacity Zhou, Xia Ye, Deping Functional Analysis Differential Geometry Metric Geometry 52A40, 52A38, 53A15, 46E30, 46E35, 28A75 Let $M_{n, m}(\mathbb{R})$ denote the space of $n\times m$ real matrices, and $\mathcal{K}_o^{n,m}$ be the set of convex bodies in $M_{n, m}(\mathbb{R})$ containing the origin. We develop a theory for the $m$th order $p$-affine capacity $C_{p,Q}(\cdot)$ for $p\in[1,n)$ and $Q\in\mathcal{K}_{o}^{1,m}$. Several equivalent definitions for the $m$th order $p$-affine capacity will be provided, and some of its fundamental properties will be proved, including for example, translation invariance and affine invariance. We also establish several inequalities related to the $m$th order $p$-affine capacity, including those comparing to the $p$-variational capacity, the volume, the $m$th order $p$-integral affine surface area, as well as the $L_p$ surface area. |
| title | On the $m$th order $p$-affine capacity |
| topic | Functional Analysis Differential Geometry Metric Geometry 52A40, 52A38, 53A15, 46E30, 46E35, 28A75 |
| url | https://arxiv.org/abs/2505.12573 |