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 $\mathcal{O}$.