In this work, we introduce and investigate a new class of sets, the kth 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 kth 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) <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\mathcal {O}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">O</mi> </math></EquationSource> </InlineEquation>: <Equation ID="Equ28"> <EquationSource Format="TEX">$ L_{\mathcal {O}}^2 - 4\pi A_{\mathcal {O}} \geqslant 4\pi |A_{\mathcal {P}_k}| + 2\pi |A_{\Omega _{\mathcal {O},k}}|, $</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <msubsup> <mi>L</mi> <mrow> <mi mathvariant="script">O</mi> </mrow> <mn>2</mn> </msubsup> <mo>-</mo> <mn>4</mn> <mi>π</mi> <msub> <mi>A</mi> <mi mathvariant="script">O</mi> </msub> <mrow> <mo>⩾</mo> <mn>4</mn> <mi>π</mi> <mo stretchy="false">|</mo> </mrow> <msub> <mi>A</mi> <msub> <mi mathvariant="script">P</mi> <mi>k</mi> </msub> </msub> <mrow> <mo stretchy="false">|</mo> <mo>+</mo> <mn>2</mn> <mi>π</mi> <mo stretchy="false">|</mo> </mrow> <msub> <mi>A</mi> <msub> <mi mathvariant="normal">Ω</mi> <mrow> <mi mathvariant="script">O</mi> <mo>,</mo> <mi>k</mi> </mrow> </msub> </msub> <mrow> <mo stretchy="false">|</mo> <mo>,</mo> </mrow> </mrow> </math></EquationSource> </Equation>where <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$L_{\mathcal {O}}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>L</mi> <mi mathvariant="script">O</mi> </msub> </math></EquationSource> </InlineEquation> denotes the length (perimeter) of <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\mathcal {O}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">O</mi> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$A_{\mathcal {O}}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>A</mi> <mi mathvariant="script">O</mi> </msub> </math></EquationSource> </InlineEquation> is the area of the region enclosed by <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\mathcal {O}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">O</mi> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$A_{\mathcal {P}_k}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>A</mi> <msub> <mi mathvariant="script">P</mi> <mi>k</mi> </msub> </msub> </math></EquationSource> </InlineEquation> is the oriented area of the associated kth Order Preserving Set <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$\mathcal {P}_k$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="script">P</mi> <mi>k</mi> </msub> </math></EquationSource> </InlineEquation>, and <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$A_{\Omega _{\mathcal {O},k}}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>A</mi> <msub> <mi mathvariant="normal">Ω</mi> <mrow> <mi mathvariant="script">O</mi> <mo>,</mo> <mi>k</mi> </mrow> </msub> </msub> </math></EquationSource> </InlineEquation> is the oriented area of the associated kth Order Midpoint Set <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$\Omega _{\mathcal {O},k}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="normal">Ω</mi> <mrow> <mi mathvariant="script">O</mi> <mo>,</mo> <mi>k</mi> </mrow> </msub> </math></EquationSource> </InlineEquation>. Moreover, we characterize the equality case: the inequality becomes an equality if and only if every equiangular circumscribed k-gon around <InlineEquation ID="IEq10"> <EquationSource Format="TEX">$\mathcal {O}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">O</mi> </math></EquationSource> </InlineEquation> is a regular k-gon with its center of mass located at the Steiner point of <InlineEquation ID="IEq11"> <EquationSource Format="TEX">$\mathcal {O}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">O</mi> </math></EquationSource> </InlineEquation>.

The kth Order Preserving Sets and isoperimetric-type inequalities for planar ovals

期刊论文
发表日期：

2026年3月7日

查看全文

Maksymilian Filip Safarewicz,
Michał Zwierzyński

摘要

In this work, we introduce and investigate a new class of sets, the kth 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 kth 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) $\mathcal {O}$ $O$ : $ L_{\mathcal {O}}^2 - 4\pi A_{\mathcal {O}} \geqslant 4\pi |A_{\mathcal {P}_k}| + 2\pi |A_{\Omega _{\mathcal {O},k}}|, $ $L_{O}^{2} - 4 π A_{O} ⩾ 4 π | A_{P_{k}} | + 2 π | A_{Ω_{O, k}} |,$ where $L_{\mathcal {O}}$ $L_{O}$ denotes the length (perimeter) of $\mathcal {O}$ $O$ , $A_{\mathcal {O}}$ $A_{O}$ is the area of the region enclosed by $\mathcal {O}$ $O$ , $A_{\mathcal {P}_k}$ $A_{P_{k}}$ is the oriented area of the associated kth Order Preserving Set $\mathcal {P}_k$ $P_{k}$ , and $A_{\Omega _{\mathcal {O},k}}$ $A_{Ω_{O, k}}$ is the oriented area of the associated kth Order Midpoint Set $\Omega _{\mathcal {O},k}$ $Ω_{O, k}$ . Moreover, we characterize the equality case: the inequality becomes an equality if and only if every equiangular circumscribed k-gon around $\mathcal {O}$ $O$ is a regular k-gon with its center of mass located at the Steiner point of $\mathcal {O}$ $O$ .