The $k$th Order Preserving Sets and Isoperimetric Type Inequalities for Planar Ovals

Abstract

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 $\mathcal{O}$.

Figures (9)

• Figure 1: An oval $\mathcal{O}$ (solid) with $p_1, p_2, p_3$ an isogonal family of points (red) and $\mathcal{O}^{\perp}$ (dashed) with $p_1^{\perp}, p_2^{\perp}, p_3^{\perp}$ (blue)
• Figure 2: On the left, oval $\mathcal{O}$ (blue, dashed) with its $3$rd Order Preserving Set (orange); on the right, $\mathcal{O}$ with its $4$th Order Preserving Set (orange)
• Figure 3: A hedgehog $\mathcal{H}$ (blue, dashed) and its $3$rd Order Preserving Set (orange)
• Figure 4: Consider an oval $\mathcal{O}$ with a support function $h(s) = 30 + \sin(2s) + \cos(3s)+ \cos(4s)$. The support line to $\mathcal{O}$ at $\mathcal{O}(s)$ is parallel to the support line to $\mathcal{P}_3$ at $\mathcal{P}_3\left(s\right)$
• Figure 5: A hedgehog $\mathcal{H}$ with a support function $h(s)=130+\sin (5s)+\sin (10s)$, a regular pentagon circumscribed about $\mathcal{H}$, and the Steiner point of $\mathcal{H}$ (centroid of the pentagon), with $v_i$, $w_i$, $\ell_i$, $t_i$, and $T_i$ indicated

Theorems & Definitions (48)

• Remark 2.1
• Definition 2.2
• Definition 2.3
• Definition 2.4
• Definition 3.1
• Example 3.2
• Definition 3.3
• Remark 3.4
• Example 3.5
• Example 3.6