The $k$th Order Preserving Sets and Isoperimetric Type Inequalities for Planar Ovals
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| Format: | Preprint |
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2025
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| _version_ | 1866908849128079360 |
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| author | Safarewicz, Maksymilian Filip Zwierzyński, Michał |
| author_facet | Safarewicz, Maksymilian Filip Zwierzyński, Michał |
| contents | In this work, we introduce and investigate a new class of sets, the \textit{$k$th Order Preserving Sets}, arising naturally from the Fourier analysis of support functions associated with hedgehogs. Specifically, we focus on sets whose support functions possess a Fourier series that preserves only terms with positive indices divisible by a fixed $k$.
We explore the geometry of the \textit{$k$th Order Midpoint Set}, defined as the set of centroids of all equiangular $k$-gons circumscribed about a given hedgehog. This set captures essential structural and symmetry-related features of the underlying geometric configuration.
We study the geometric properties of such sets and, in particular, establish an isoperimetric-type inequality relating the perimeter and area of a region bounded by a simple smooth convex closed curve (an oval) $\mathcal{O}$: \[ L_{\mathcal{O}}^2 - 4πA_{\mathcal{O}} \geqslant 4π|A_{\mathcal{P}_k}| + 2π|A_{Ω_{\mathcal{O},k}}|, \] where $L_{\mathcal{O}}$ denotes the length (perimeter) of $\mathcal{O}$, $A_{\mathcal{O}}$ is the area of the region enclosed by $\mathcal{O}$, $A_{\mathcal{P}_k}$ is the oriented area of the associated $k$th Order Preserving Set $\mathcal{P}_k$, and $A_{Ω_{\mathcal{O},k}}$ is the oriented area of the associated $k$th Order Midpoint Set $Ω_{\mathcal{O},k}$. Moreover, we characterize the equality case: the inequality becomes an equality if and only if every equiangular circumscribed $k$-gon around $\mathcal{O}$ is a~regular $k$-gon with its center of mass located at the Steiner point of $\mathcal{O}$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2505_08017 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The $k$th Order Preserving Sets and Isoperimetric Type Inequalities for Planar Ovals Safarewicz, Maksymilian Filip Zwierzyński, Michał Differential Geometry Primary: 52A38, 52A40, 53A04. Secondary: 52A10, 58K70 In this work, we introduce and investigate a new class of sets, the \textit{$k$th Order Preserving Sets}, arising naturally from the Fourier analysis of support functions associated with hedgehogs. Specifically, we focus on sets whose support functions possess a Fourier series that preserves only terms with positive indices divisible by a fixed $k$. We explore the geometry of the \textit{$k$th Order Midpoint Set}, defined as the set of centroids of all equiangular $k$-gons circumscribed about a given hedgehog. This set captures essential structural and symmetry-related features of the underlying geometric configuration. We study the geometric properties of such sets and, in particular, establish an isoperimetric-type inequality relating the perimeter and area of a region bounded by a simple smooth convex closed curve (an oval) $\mathcal{O}$: \[ L_{\mathcal{O}}^2 - 4πA_{\mathcal{O}} \geqslant 4π|A_{\mathcal{P}_k}| + 2π|A_{Ω_{\mathcal{O},k}}|, \] where $L_{\mathcal{O}}$ denotes the length (perimeter) of $\mathcal{O}$, $A_{\mathcal{O}}$ is the area of the region enclosed by $\mathcal{O}$, $A_{\mathcal{P}_k}$ is the oriented area of the associated $k$th Order Preserving Set $\mathcal{P}_k$, and $A_{Ω_{\mathcal{O},k}}$ is the oriented area of the associated $k$th Order Midpoint Set $Ω_{\mathcal{O},k}$. Moreover, we characterize the equality case: the inequality becomes an equality if and only if every equiangular circumscribed $k$-gon around $\mathcal{O}$ is a~regular $k$-gon with its center of mass located at the Steiner point of $\mathcal{O}$. |
| title | The $k$th Order Preserving Sets and Isoperimetric Type Inequalities for Planar Ovals |
| topic | Differential Geometry Primary: 52A38, 52A40, 53A04. Secondary: 52A10, 58K70 |
| url | https://arxiv.org/abs/2505.08017 |