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| Format: | Preprint |
| Publié: |
2024
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| Accès en ligne: | https://arxiv.org/abs/2411.01631 |
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| _version_ | 1866915003567702016 |
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| author | Besau, Florian Werner, Elisabeth M. |
| author_facet | Besau, Florian Werner, Elisabeth M. |
| contents | We explore analogs of classical centro-affine invariant isoperimetric inequalities, such as the Blaschke--Santaló inequality and the $L_p$-affine isoperimetric inequalities, for convex bodies in spherical space. Specifically, we establish an isoperimetric inequality for the floating area and prove a stability result based on the spherical volume difference.
The floating area has previously been studied as a natural extension of classical affine surface area to non-Euclidean convex bodies in spaces of constant curvature. In this work, we introduce the $L_p$-floating areas for spherical convex bodies, extending Lutwak's centro-affine invariant family of $L_p$-affine surface area measures from Euclidean geometry. We prove a duality formula, monotonicity properties, and isoperimetric inequalities associated with this new family of curvature measures for spherical convex bodies.
Additionally, we propose a novel curvature entropy functional for spherical convex bodies, based on the $L_p$-floating area, and establish a corresponding dual isoperimetric inequality.
Finally, we extend our spherical notions to space forms with non-negative constant curvature in two distinct ways. One extension asymptotically connects with centro-affine geometry on convex bodies as curvature approaches zero, while the other converges with Euclidean geometry. Notably, our newly introduced curvature entropy for spherical convex bodies emerges as a natural counterpart to both the centro-affine entropy and the Gaussian entropy of convex bodies in Euclidean space. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_01631 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The $L_p$-floating area, curvature entropy, and isoperimetric inequalities on the sphere Besau, Florian Werner, Elisabeth M. Metric Geometry Differential Geometry Primary 52A55, Secondary 28A75, 52A20, 53A35 We explore analogs of classical centro-affine invariant isoperimetric inequalities, such as the Blaschke--Santaló inequality and the $L_p$-affine isoperimetric inequalities, for convex bodies in spherical space. Specifically, we establish an isoperimetric inequality for the floating area and prove a stability result based on the spherical volume difference. The floating area has previously been studied as a natural extension of classical affine surface area to non-Euclidean convex bodies in spaces of constant curvature. In this work, we introduce the $L_p$-floating areas for spherical convex bodies, extending Lutwak's centro-affine invariant family of $L_p$-affine surface area measures from Euclidean geometry. We prove a duality formula, monotonicity properties, and isoperimetric inequalities associated with this new family of curvature measures for spherical convex bodies. Additionally, we propose a novel curvature entropy functional for spherical convex bodies, based on the $L_p$-floating area, and establish a corresponding dual isoperimetric inequality. Finally, we extend our spherical notions to space forms with non-negative constant curvature in two distinct ways. One extension asymptotically connects with centro-affine geometry on convex bodies as curvature approaches zero, while the other converges with Euclidean geometry. Notably, our newly introduced curvature entropy for spherical convex bodies emerges as a natural counterpart to both the centro-affine entropy and the Gaussian entropy of convex bodies in Euclidean space. |
| title | The $L_p$-floating area, curvature entropy, and isoperimetric inequalities on the sphere |
| topic | Metric Geometry Differential Geometry Primary 52A55, Secondary 28A75, 52A20, 53A35 |
| url | https://arxiv.org/abs/2411.01631 |