On general versions of the Petty projection inequality
Fuente:
arXiv
Guardado en:
| Autor principal: | |
|---|---|
| Formato: | Preprint |
| Publicado: |
2025
|
| Materias: | |
| Acceso en línea: | |
| Etiquetas: |
Agregar Etiqueta
Sin Etiquetas, Sea el primero en etiquetar este registro!
|
| _version_ | 1866908506912718848 |
|---|---|
| author | Sola, Francisco Marín |
| author_facet | Sola, Francisco Marín |
| contents | The classical Petty projection inequality is an affine isoperimetric inequality which constitutes a cornerstone in the affine geometry of convex bodies. By extending the polar projection body to an inter-dimensional operator, Petty's inequality was generalized to the so-called $(L_p,Q)$ setting, where $Q$ is an $m$-dimensional compact convex set. In this work, we further extend the $(L_p,Q)$ Petty projection inequality to the broader realm of rotationally invariant measures with concavity properties, namely, those with $γ$-concave density (for $γ\geq-1/nm$). Moreover, when $p=1$, and motivated by a contemporary empirical reinterpretation of Petty's result, we explore empirical analogues of this inequality. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_00949 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On general versions of the Petty projection inequality Sola, Francisco Marín Metric Geometry Functional Analysis The classical Petty projection inequality is an affine isoperimetric inequality which constitutes a cornerstone in the affine geometry of convex bodies. By extending the polar projection body to an inter-dimensional operator, Petty's inequality was generalized to the so-called $(L_p,Q)$ setting, where $Q$ is an $m$-dimensional compact convex set. In this work, we further extend the $(L_p,Q)$ Petty projection inequality to the broader realm of rotationally invariant measures with concavity properties, namely, those with $γ$-concave density (for $γ\geq-1/nm$). Moreover, when $p=1$, and motivated by a contemporary empirical reinterpretation of Petty's result, we explore empirical analogues of this inequality. |
| title | On general versions of the Petty projection inequality |
| topic | Metric Geometry Functional Analysis |
| url | https://arxiv.org/abs/2503.00949 |