Theories and Inequalities on the Satellite System

Abstract

We define the satellite system without any central, the satellite system with a central, and the satellite system of single point with a central. For the satellite system without any central 𝑆{Γ,Γ_𝑁}, we establish the inequality: ∮_Γ𝑑𝑠∮_Γ𝑁|𝑃𝑄|𝑑𝑠_𝑄≤(𝑁|Γ|³/4𝜋²)sin(2𝜋/𝑁). For the satellite system with a central 𝑆{𝑂,Γ,Γ_𝑁}, we establish the following inequality under the proper hypothesis: ∮_Γ𝑀^𝑁[𝑡](𝑟)𝑑𝑠≤(|Γ|²/2𝜋)cos(𝜋/𝑁)(∀𝑡∈]−∞,−2]). As an application, we get the inequalities ∮_Γ𝜌_𝑃𝑑𝑠≥∮_Γ𝑟_𝑃𝑑𝑠≥2|𝐷(Γ)|, for the satellite system of single point with a central 𝑆{𝑂,Γ,𝑃}. For these results, there are generalized backgrounds in the fields of differential geometry and space science.

1. Introduction and the Main Results

In this paper, we will use the following symbols: 𝑅 and 𝑍 to denote the set of real numbers and the set of integers, respectively, 𝑎=(𝑎₁,𝑎₂,…,𝑎_𝑛), and 𝑅^𝑛={𝑎|𝑎_𝑖∈𝑅,1≤𝑖≤𝑛}[1, 2].

Definition 1.1. Let Γ=Γ(𝐴𝐵)∶𝑟=𝑟(𝑡)=𝑥(𝑡)𝑖⃗+𝑦(𝑡)𝑗⃗+𝑧(𝑡)𝑘⃗(𝑎≤𝑡≤𝑏) be a smooth curve in three-space 𝑅³. 𝐴=𝑟(𝑎) and 𝐵=𝑟(𝑏) are called the initial point and terminal point of Γ, respectively. If 𝑎<𝑡₁<𝑏, 𝑎≤𝑡₂≤𝑏, 𝑡₁≠𝑡₂, and 𝑟(𝑡₁)=𝑟(𝑡₂), then 𝑟(𝑡₁) is called a coincident point of Γ; smooth curve without any coincident point is called a simple curve; If a simple curve Γ satisfies 𝐴=𝐵, then Γ is said be a simple closed curve. Especially, we say Γ is a Jordan closed curve if Γ is a plane curve in 𝑅³. For a Jordan closed curve Γ, 𝐷(Γ) denotes the closed region bounded by Γ and its area is written as |𝐷(Γ)|, |Γ| denotes the length of Γ, and we have the following Jordan curve theorem.

Theorem 1.2 (Jordan Curve [3]). An arbitrary Jordan closed curve must divide a plane into two parts, where one part is bounded, and the other is unbounded. The bounded part is called interior and another is outside of the Jordan closed curve.

Definition 1.3 (see [4, 5]). Assume that Γ(𝐴𝐵) is a smooth space curve with end points 𝐴 and 𝐵, 𝑃 is a fixed point in 𝑅³, and 𝑄∈Γ(𝐴𝐵), where 𝑄 is a moving point, then the trail of the line segment 𝑃𝑄 is called a bounded cone surface, written as Ω[𝑃,Γ(𝐴𝐵)]. We Also say 𝑃𝑄 is the generating line, 𝑃 is the vertex, and Γ(𝐴𝐵) is the generating curve of the bounded cone surface. |Ω[𝑃,Γ(𝐴𝐵)]| denotes the area of Ω[𝑃,Γ(𝐴𝐵)].
If Γ is a given smooth curve in 𝑅³, then its length is written as |Γ|; Γ(𝐴𝐵) denotes the smooth curve segment with end points 𝐴 and 𝐵, and Γ(𝐴𝐵)⊂Γ.

Definition 1.4. Let Γ be a fixed smooth closed curve in 𝑅³, the 𝑁-polygon Γ_𝑁∶𝐴₁𝐴₂⋯𝐴_𝑁𝐴₁(𝑁≥3) be inscribed in Γ, and, 𝐴_𝑖∈Γ(𝐴_𝑖−1𝐴_𝑖+1)(𝑖=1,2,…,𝑁). If all vertices∶𝐴₁,𝐴₂,…,𝐴_𝑁 of Γ_𝑁 move continuously along Γ and keep 𝑙_𝑖∶=|Γ(𝐴_𝑖𝐴_𝑖+1)| invariable, where 0<𝑙_𝑖<|Γ|/2,𝑖=1,2,…,𝑁, we say that the set 𝑆{Γ,Γ_𝑁}∶={Γ,Γ_𝑁} is the satellite system without any central in three-space 𝑅³.
For 𝑆{Γ,Γ_𝑁}, we write the set of the vertices of Γ_𝑁 as 𝑉(Γ_𝑁)∶={𝐴₁,𝐴₂,…,𝐴_𝑁} and define 𝐴_𝑖=𝐴_𝑗⇔𝑖≡𝑗(mod𝑁), for all 𝑖,𝑗∈𝑍 [6].

Definition 1.5. For 𝑆{Γ,Γ_𝑁}, if the following statements are valid:(i)Γis a fixed closed Jordan curve in 𝑅³,(ii)𝑂 is a fixed point in the interior of Γ, and(iii)Γ_𝑁 is always a Jordan closed curve and Ω[𝑂,Γ_𝑁]=𝐷(Γ_𝑁) as all vertices of Γ_𝑁 move continuously along Γ,then we say the set 𝑆{𝑂,Γ,Γ_𝑁}:={𝑂,Γ,Γ_𝑁} is the satellite system with a central and 𝑂 is its central.
For 𝑆{𝑂,Γ,Γ_𝑁}, 𝑟_𝑖∶=Dis(𝑂,𝐴_𝑖𝐴_𝑖+1) denotes the distance from 𝑂 to the line 𝐴_𝑖𝐴_𝑖+1, 𝑖=1,2,…,𝑁. And we write 𝑟∶=(𝑟₁,𝑟₂,…,𝑟_𝑁).

Remark 1.6. The 𝑆{𝑂,Γ,Γ_𝑁} may be explained as follows: the point 𝑂 denotes the central of earth, Γ denotes the trajectory on which satellites move, and the vertexes of Γ_𝑁 are viewed as 𝑁 satellites moving on the same trajectory Γ. In order to avoid hitting, they must move by same curve velocity, that is, 𝑙_𝑖∶=|Γ(𝐴_𝑖𝐴_𝑖+1)| is invariable and 0<𝑙_𝑖<|Γ|/2,𝑖=1,2,…,𝑁.

Definition 1.7. The tth power mean of the positive real numbers sets 𝑎=(𝑎₁,𝑎₂,…,𝑎_𝑁)(𝑁≥2), written as 𝑀^𝑁[𝑡](𝑎), is defined by [7–15]: 𝑀^𝑁[𝑡](𝑎)∶=⎧⎪⎪⎪⎨⎪⎪⎪⎩1𝑁^𝑁_𝑖=1𝑎^𝑡𝑖^1/𝑡,𝑡∈𝑅,0<|𝑡|<+∞_𝑁⎷_𝑁_𝑖=1𝑎_𝑖,𝑡=0.(1.1)

Definition 1.8 (see [1, 7, 10, 11, 14]). Let Γ be a smooth curve in 𝑅³. For Riemann integrable function 𝑓∶Γ→𝑅, 𝑀(𝑓,Γ)∶=(1/|Γ|)∫_Γ𝑓𝑑𝑠 is called function means of 𝑓 on Γ. For the function 𝑓∶Γ→]0,∞[ and the real number 𝑡, if 𝑓^𝑡(𝑡∈𝑅,𝑡≠0) and ln𝑓 are Riemann integrable on Γ, then 𝑀^[𝑡](𝑓,Γ)∶=⎧⎪⎪⎨⎪⎪⎩1||Γ||_Γexp1𝑓^𝑡𝑑𝑠^1/𝑡,𝑡∈𝑅,𝑡≠0||Γ||_Γln𝑓𝑑𝑠,𝑡=0(1.2) is called the tth power means of 𝑓 on Γ.

Now, we give our main results as follow.

Theorem 1.9. Let 𝑆{Γ,Γ_𝑁} be a satellite system without any central. For any 𝑃∈𝑉(Γ_𝑁)∶={𝐴₁,𝐴₂,…,𝐴_𝑁}, we have inequality _Γ𝑑𝑠_Γ𝑁||𝑃𝑄||𝑑𝑠_𝑄≤𝑁||Γ||³4𝜋²sin2𝜋𝑁,(1.3) that is, 𝑀||Ω𝑃,Γ_𝑁||,Γ≤𝑁||Γ||²8𝜋²sin2𝜋𝑁,(1.4) where 𝑃𝑄 is the generating line of the bounded cone surface Ω[𝑃,Γ_𝑁]. A sufficient condition of equalities in (1.3) and (1.4) is that Γ is a circle and Γ_𝑁 is always a regular polygon with 𝑁 sides.

Theorem 1.10. Let 𝑆{𝑂,Γ,Γ_𝑁} be a satellite system with a central. If 𝑁=3 or 𝑁≥4, 0<𝑙_𝑖≤|Γ|/4,𝑖=1,2,…,𝑁, we have inequality, for any the real number 𝑡≤−2, _Γ𝑀^𝑁[𝑡](𝑟)𝑑𝑠≤||Γ||²2𝜋cos𝑁𝜋,(1.5) that is, 𝑀𝑀^𝑁[𝑡](𝑟),Γ≤||Γ||2𝜋cos𝑁𝜋.(1.6) The equalities of (1.5) and (1.6) occur if and only if Γ is a circle, 𝑂 is the central of the circle, and Γ_𝑁 is a regular polygon with sides 𝑁.

In Section 5, we will give some applications of these results and theories.

2. Preliminaries

Lemma 2.1. For the quadrilateral 𝐴𝐵𝐶𝐷𝐴 in 𝑅³, writing 𝑎∶=|𝐴𝐵|,𝑏∶=|𝐵𝐶|,𝑐∶=|𝐶𝐷|,𝑑∶=|𝐷𝐴|, we obtain ||Δ𝐵𝐴𝐷||+||Δ𝐵𝐶𝐷||≤14(𝑎+𝑑)²−(𝑏−𝑐)²(𝑏+𝑐)²−(𝑎−𝑑)²,(2.1) with equality if and only if ∠𝐵𝐴𝐷+∠𝐵𝐶𝐷=𝜋. If the quadrilateral 𝐴𝐵𝐶𝐷𝐴 is convex and in plane, equality holds if and only if the quadrilateral 𝐴𝐵𝐶𝐷𝐴 is inscribed in a circle.

Proof. Write 𝑠∶=|Δ𝐵𝐴𝐷|+|Δ𝐵𝐶𝐷|,∠𝐵𝐴𝐷=𝛼,∠𝐵𝐶𝐷=𝛽. By the area formula of triangle and cosine theorem, we get 𝑠=1𝑎2(𝑎𝑑sin𝛼+𝑏𝑐sin𝛽),(2.2)₂+𝑑²−2𝑎𝑑cos𝛼=𝑏²+𝑐²−2𝑏𝑐cos𝛽,0≤𝛼,𝛽≤𝜋.(2.3) Consider 𝛽=𝛽(𝛼) as the implicit function with respect to 𝛼. Finding the derivatives of the two sides of (2.2) and (2.3), respectively, we have 𝑑𝑠𝑑𝛼=12𝑎𝑑cos𝛼+𝑏𝑐cos𝛽𝑑𝑑𝛼𝛽,𝑑𝑑𝛼𝛽=𝑎𝑏𝑑𝑐ssiinn𝛽𝛼≥0.(2.4) Therefore, (𝑑𝑠/𝑑𝛼)=(1/2)𝑎𝑑sin𝛼(cot𝛼+cot𝛽) and 𝛽=𝛽(𝛼) is increasing with respect to 𝛼. Since ||⟹⟹𝑏−𝑐(||||𝑎𝑏≤+−||𝑑𝑐𝐵||)𝐷₂≤||𝑎≤+𝑎𝑑+,𝑑,||𝑎−||𝑎𝑑−||≤𝑑||𝑏≤+||𝐵𝑐,𝐷||≤𝑏+𝑐,⟹−2(𝑎𝑑+−𝑏(𝑏𝑐)−≤𝑐𝑎)²₂≥0,(𝑎−𝑑)²−(𝑏+𝑐)²≤0,⟹||||𝑎+𝑑²−𝑏²−𝑐²≤2(𝑎𝑑+𝑏𝑐),₂+𝑑²−𝑏²−𝑐²2(𝑎𝑑+𝑏𝑐)||||≤1,(2.5) we have 𝑑𝑠𝑑𝛼=0⟺𝛼+𝛽=𝜋,𝑎²⟺𝛼=𝛼₀+𝑑²−2𝑎𝑑cos𝛼=𝑏²+𝑐²−2𝑏𝑐cos𝛽,∶=arccos𝑎²+𝑑²−𝑏²−𝑐²2(𝑎𝑑+𝑏𝑐),𝛽=𝛽𝛼⁰=𝜋−𝛼₀0≤𝛼<𝛼₀,⟹0≤𝛽=𝛽(𝛼)<𝛽𝛼₀=𝜋−𝛼₀⟹0≤𝛽<𝜋−𝛼⟹𝑑𝑠𝛼𝑑𝛼>0,₀<𝛼≤𝜋⟹𝜋≥𝛽=𝛽(𝛼)>𝛽𝛼₀=𝜋−𝛼₀⟹𝜋≥𝛽>𝜋−𝛼⟹𝑑𝑠𝑑𝛼<0.(2.6) Hence 𝑠 is max if and only if 𝛼+𝛽=𝜋. When 𝑠 is max, 𝛽=𝜋−𝛼,𝛼=𝛼₀. Thus, 𝑠≤12𝑎𝑑sin𝛼⁰+𝑏𝑐sin𝛼₀=12(𝑎𝑑+𝑏𝑐)1−𝑎²+𝑑²−𝑏²−𝑐²2(𝑎𝑑+𝑏𝑐)²=14(𝑎+𝑑)²−(𝑏−𝑐)²(𝑏+𝑐)²−(𝑎−𝑑)².(2.7) In other words, (2.1) holds. Equality holds if and only if 𝛼+𝛽=𝜋⇔∠𝐵𝐴𝐷+∠𝐵𝐶𝐷=𝜋. This completes the proof.

Lemma 2.2. Let 𝑆{Γ,Γ_𝑁} be a satellite system without any central. For any 𝑃∈𝑉(Γ_𝑁), we have inequality: 12_Γ𝑁||𝑃𝑄||𝑑𝑠_𝑄=||Ω𝑃,Γ_𝑁||≤||Γ^𝑁||²4𝑁cot𝑁𝜋,(2.8) where 𝑃𝑄 is the generating line of the bounded cone surface Ω[𝑃,Γ_𝑁]. The second equality occurs if Γ_𝑁 is a regular polygon with sides 𝑁.

Proof. By the definition and geometric meaning of curve integral, we obtain 12_Γ𝑁||𝑃𝑄||𝑑𝑠_𝑄=||Ω𝑃,Γ_𝑁||.(2.9) Now we prove the inequality in (2.8).
If 𝑁=3, (2.8) is known; if 𝑁=4, by Lemma 2.1, (2.8) holds. In the following, we suppose that 𝑁≥5. First, fix the value of |Γ_𝑁|. Without loss of generality, we set 𝑃=𝐴₁. By the theory of differential geometry, we know a bounded cone surface is a developable surface, which implies that Ω[𝑃,Γ_𝑁] can be developed into a bounded cone surface Ω[𝐴^∗1,Γ^∗𝑁] in the plane, where Γ^∗𝑁∶𝐴^∗1𝐴^∗2⋯𝐴^∗𝑁𝐴^∗1(𝑃=𝐴₁=𝐴^∗1) is the developed graph of Γ_𝑁 in the plane and it is a polygon with sides 𝑁 (may not be a closed Jordan curve) and satisfies Δ𝐴^∗𝑖−1𝐴^∗1𝐴^∗𝑖∩||ΩΔ𝐴𝑃,^∗𝑖Γ𝐴^∗1_𝑁𝐴^∗𝑖+1=𝐴^∗1𝐴^∗𝑖(𝑖=3,4,…,𝑁−1),||Γ^∗𝑁||=||Γ_𝑁||,||=||Ω𝐴^∗1,Γ^∗𝑁||=^𝑁_𝑖=3||Δ𝐴^∗𝑖−1𝐴^∗1𝐴^∗𝑖||.(2.10) When |Ω[𝐴^∗1,Γ^∗𝑁]| is max, for any 𝑖∈{2,3,…,𝑁−1}, the quadrilateral 𝐴^∗𝑖−1𝐴^∗𝑖𝐴^∗𝑖+1𝐴^∗𝑖+2𝐴^∗𝑖−1 must be convex and the polygon with 5 sides 𝐴^∗1𝐴^∗𝑖−1𝐴^∗𝑖𝐴^∗𝑖+1𝐴^∗𝑖+2𝐴^∗1(𝐴^∗𝑁+1=𝐴^∗1) must be also convex by the plane geometry. Now, we prove the four points 𝐴^∗𝑖−1,𝐴^∗𝑖,𝐴^∗𝑖+1 and 𝐴^∗𝑖+2 are on a common circle. Otherwise, there exists some 𝑖∈{2,3,…,𝑁−1} such that 𝐴^∗𝑖−1,𝐴^∗𝑖,𝐴^∗𝑖+1,𝐴^∗𝑖+2 are not on a common circle. Therefore, fix the point 𝐴^∗𝑗(𝑗∈{1,2,…,𝑁}−{𝑖,𝑖+1}) and modify 𝐴^∗𝑖,𝐴^∗𝑖+1 to 𝐴^𝑖∗∗,𝐴^∗∗𝑖+1 such that 𝐴^∗𝑖−1,𝐴^𝑖∗∗,𝐴^∗∗𝑖+1,𝐴^∗𝑖+2 are on a common circle and |||𝐴_∗𝑖−1𝐴^𝑖∗∗|||=|||𝐴_∗𝑖−1𝐴^∗𝑖|||,|||𝐴_𝑖∗∗𝐴^∗∗𝑖+1|||=|||𝐴_∗𝑖𝐴^∗𝑖+1|||,|||𝐴_∗∗𝑖+1𝐴^∗𝑖+2|||=|||𝐴_∗𝑖+1𝐴^∗𝑖+2|||.(2.11) Hence, we obtain a new polygon with 𝑁 sides and write Γ^𝑁∗∗∶𝐴^∗1𝐴^∗2⋯𝐴^∗𝑖−1𝐴^𝑖∗∗𝐴^∗∗𝑖+1𝐴^∗𝑖+2⋯𝐴^∗𝑁𝐴^∗1.(2.12) It follows that ||Ω||Ω𝐴𝐴^∗1_∗1,Γ^∗𝑁||=||Ω𝐴^∗1,𝐴^∗1𝐴^∗2⋯𝐴^∗𝑖−1𝐴^∗𝑖+2⋯𝐴^∗𝑁𝐴^∗1||+||𝐷𝐴^∗𝑖−1𝐴^∗𝑖𝐴^∗𝑖+1𝐴^∗𝑖+2𝐴^∗𝑖−1||,,Γ^𝑁∗∗||=||Ω𝐴^∗1,𝐴^∗1𝐴^∗2⋯𝐴^∗𝑖−1𝐴^∗𝑖+2⋯𝐴^∗𝑁𝐴^∗1||+||𝐷𝐴^∗𝑖−1𝐴^𝑖∗∗𝐴^∗∗𝑖+1𝐴^∗𝑖+2𝐴^∗𝑖−1||.(2.13) By Lemma 2.1, we have ||Ω𝐴^∗1,Γ^𝑁∗∗||−||Ω𝐴^∗1,Γ^∗𝑁||=||𝐷𝐴^∗𝑖−1𝐴^𝑖∗∗𝐴^∗∗𝑖+1𝐴^∗𝑖+2𝐴^∗𝑖−1||−||𝐷𝐴^∗𝑖−1𝐴^∗𝑖𝐴^∗𝑖+1𝐴^∗𝑖+2𝐴^∗𝑖−1||>0.(2.14) This contradicts the greatest |Ω[𝐴^∗1,Γ^∗𝑁]|. Since three points confirm a unique circle and for any 𝑖∈{2,3,…,𝑁−1}, we know 𝐴^∗1,𝐴^∗2,…,𝐴^∗𝑁−1,𝐴^∗𝑁 are on a common circle if |Ω[𝐴^∗1,Γ^∗𝑁]| is greatest, and since Δ𝐴^∗𝑖−1𝐴^∗1𝐴^∗𝑖∩Δ𝐴^∗𝑖𝐴^∗1𝐴^∗𝑖+1=𝐴^∗1𝐴^∗𝑖,𝑖=3,4,…,𝑁−1,(2.15)Γ^∗𝑁 is inscribed in a circle and |Ω[𝑃,Γ_𝑁]|=|Ω[𝐴^∗1,Γ^∗𝑁]|=|𝐷(Γ^∗𝑁)|. Otherwise, there exists 𝑘∈{2,3,…,𝑁−1} such that Δ𝐴^∗𝑘−1𝐴^∗1𝐴^∗𝑘∩Δ𝐴^∗𝑘𝐴^∗1𝐴^∗𝑘+1≠𝐴^∗1𝐴^∗𝑘. When the perimeter of a circle is a fixed value, the area of regular polygon with 𝑁 sides is greatest in all N-polygons inscribed in the circle [16]. Therefore, Γ^∗𝑁 is a regular polygon with 𝑁 sides if |Ω[𝐴^∗1,Γ^∗𝑁]| is greatest.
Now let Γ^∗𝑁 be a regular polygon with 𝑁 sides. By the plane geometry, we know||Ω𝐴^∗1,Γ^∗𝑁||=||𝐷Γ^∗𝑁||=||Γ^∗𝑁||²4𝑁cot𝑁𝜋=||Γ^𝑁||²4𝑁cot𝑁𝜋.(2.16) Hence, the inequality holds in (2.8). Equality occurs if Γ^∗𝑁 is a regular 𝑁-polygon. This completes the proof.

Lemma 2.3. For the quadrilateral 𝐴𝐵𝐶𝐷𝐴 in 𝑅³, we have ||𝐴𝐶||²+||𝐵𝐷||²≤||𝐵𝐶||²+||𝐷𝐴||²+2||𝐴𝐵||⋅||𝐶𝐷||,(2.17) with equality if and only if ∠(𝐴𝐵,𝐶𝐷)=𝜋.

Proof. Write (𝐴𝐵,𝐵𝐶,𝐶𝐷,𝐷𝐴,𝐴𝐶,𝐵𝐷)=(𝑎,𝑏,𝑐,𝑑,𝑒,𝑓), 𝑎⋅𝑏∶=⟨𝑎,𝑏⟩ denotes the inner product of vectors 𝑎 and 𝑏, especially, 𝑎²=𝑎⋅𝑎=|𝑎|². Thus, (2.17) is expressed as 𝑒²+𝑓²≤𝑏²+𝑑²+2|𝑎|⋅|𝑐|.(2.18) Since ⎧⎪⎪⎨⎪⎪⎩𝑎𝑐𝑏𝑑++++𝑑𝑐𝑏𝑎====𝑓−𝑒−𝑒𝑓⟹⎧⎪⎪⎨⎪⎪⎩2𝑎⋅𝑏=𝑒²−𝑎²−𝑏²2𝑐⋅𝑑=𝑒²−𝑐²−𝑑²2𝑏⋅𝑐=𝑓²−𝑏²−𝑐²2𝑑⋅𝑎=𝑓²−𝑑²−𝑎²(2.19) and ∑𝑎∶=𝑎+𝑏+𝑐+𝑑=0, which implies that 0==−𝑎₂𝑎²=𝑎²+𝑏²+𝑐²+𝑑²+2(𝑎⋅𝑏+𝑐⋅𝑑+𝑏⋅𝑐+𝑑⋅𝑎)+2(𝑎⋅𝑐+𝑏⋅𝑑)=−2𝑎₂+𝑏²+𝑐²+𝑑²+2𝑒²+𝑓²+2(𝑎⋅𝑐+𝑏⋅𝑑)+𝑏²+𝑐²+𝑑²+2𝑒²+𝑓²+(𝑎+𝑐)²+(𝑏+𝑑)²=−2𝑎²+𝑏²+𝑐²+𝑑²+2𝑒²+𝑓²+2(𝑎+𝑐)²=−2𝑏²+𝑑²+2𝑒²+𝑓²+4𝑎⋅𝑐,(2.20) it follows that 𝑒²+𝑓²=𝑏²+𝑑²−2𝑎⋅𝑐≤𝑏²+𝑑²+2|𝑎|⋅|𝑐|.(2.21)(2.18) is proved. Equality of (2.18) holds if and only if ∠(𝐴𝐵,𝐶𝐷)=∠(𝑎,𝑐)=𝜋. This completes the proof.

Lemma 2.4. Let Γ_𝑁(𝑁≥4) be a polygon with sides 𝑁 in 𝑅³. Setting 𝑆_𝑘∶=sin𝑘𝜋𝑁,𝐿^𝑘∶=𝑁(2𝑆_𝑘)²^𝑁−1_𝑖=1||𝐴_𝑖𝐴_𝑖+𝑘||²,𝑘=1,2,…,𝑁−1,(2.22) we get the following inequality: 𝐿_𝑘≤𝑆¹𝑆_𝑘²𝐿₁+𝑆^𝑘−1𝑆_𝑘+12𝑆^2𝑘𝐿_𝑘+1+𝐿_𝑘−1,(2.23) for 2≤𝑘≤𝑁−2. A sufficient condition of equality is that Γ_𝑁 is a plane regular 𝑁-polygon in 𝑅³.

Proof. Consider the quadrilateral 𝐴_𝑖−1𝐴_𝑖𝐴_{𝑖−1+𝑘}𝐴_𝑖+𝑘𝐴_𝑖−1. From 𝑆^−1𝑘−1||𝐴_𝑖𝐴_{𝑖−1+𝑘}||−𝑆^−1𝑘+1||𝐴_𝑖−1𝐴_𝑖+𝑘||²≥0,(2.24) we obtain 2||𝐴_𝑖𝐴_{𝑖−1+𝑘}||⋅||𝐴_𝑖−1𝐴_𝑖+𝑘||≤𝑆^𝑘+1𝑆_𝑘−1||𝐴_𝑖𝐴_{𝑖−1+𝑘}||²+𝑆^𝑘−1𝑆_𝑘+1||𝐴_𝑖−1𝐴_𝑖+𝑘||².(2.25) It follows from Lemma 2.3 and (2.25) that ||𝐴_𝑖𝐴_𝑖+𝑘||²+||𝐴_𝑖−1𝐴_{𝑖−1+𝑘}||²≤≤||||𝐴𝐴_{𝑖−1𝑖−1}𝐴_𝑖||²+||𝐴_{𝑖−1+𝑘}𝐴_𝑖+𝑘||²+2||𝐴_𝑖𝐴_{𝑖−1+𝑘}||||𝐴_𝑖−1𝐴_𝑖+𝑘||𝐴_𝑖||²+||𝐴_{𝑖−1+𝑘}𝐴_𝑖+𝑘||²+𝑆^𝑘+1𝑆_𝑘−1||𝐴_𝑖𝐴_{𝑖−1+𝑘}||²+𝑆^𝑘−1𝑆_𝑘+1||𝐴_𝑖−1𝐴_𝑖+𝑘||²,(2.26) which implies that ^𝑁_𝑖=1||𝐴^𝑖≤_𝑁𝐴_𝑖+𝑘||²+||𝐴_𝑖−1𝐴_{𝑖−1+𝑘}||²_𝑖=1||𝐴^𝑖−1𝐴_𝑖||²+||𝐴_{𝑖−1+𝑘}𝐴_𝑖+𝑘||²+𝑆^𝑘+1𝑆_𝑘−1||𝐴_𝑖𝐴_{𝑖−1+𝑘}||²+𝑆^𝑘−1𝑆_𝑘+1||𝐴_𝑖−1𝐴_𝑖+𝑘||².(2.27) Since ^𝑁_𝑖=1||𝐴_𝑖𝐴_𝑖+𝑘||²=^𝑁_𝑖=1||𝐴_𝑖−1𝐴_{𝑖−1+𝑘}||²=4𝑁𝑆^2𝑘𝐿_𝑘,_𝑁_𝑖=1||𝐴_𝑖−1𝐴_𝑖||²=^𝑁_𝑖=1||𝐴_{𝑖−1+𝑘}𝐴_𝑖+𝑘||²=4𝑁𝑆²¹𝐿₁,_𝑁_𝑖=1||𝐴_𝑖𝐴_{𝑖−1+𝑘}||²=4𝑁𝑆^2𝑘−1𝐿_𝑘−1,^𝑁_𝑖=1||𝐴_𝑖−1𝐴_𝑖+𝑘||²=4𝑁𝑆^2𝑘+1𝐿_𝑘+1,(2.28) the inequality (2.27) is equivalent to 8𝑁𝑆^2𝑘𝐿_𝑘≤8𝑁𝑆²¹𝐿₁+4𝑁𝑆_𝑘+1𝑆_𝑘−1𝐿_𝑘−1+4𝑁𝑆_𝑘−1𝑆_𝑘+1𝐿_𝑘+1⟺𝐿_𝑘≤𝑆¹𝑆_𝑘²𝐿₁+𝑆^𝑘−1𝑆_𝑘+12𝑆^2𝑘𝐿_𝑘+1+𝐿_𝑘−1.(2.29) Inequality (2.23) is proved. From this proof and Lemma 2.3, we know that a sufficient condition of equality is that Γ_𝑁 is a regular polygon with 𝑁 sides in 𝑅³.

Remark 2.5. A sufficient condition of equality of (2.23) is that Γ_𝑁 is a regular polygon with 𝑁 sides in 𝑅³. This condition is not necessary. For example, when 𝑁=4, the equality holds in (2.23) if and only if Γ₄ is a parallelogram in 𝑅³.

Remark 2.6. If Γ_𝑁 is a regular 𝑁-polygon, 𝐿_𝑘 defined by Lemma 2.4 is equal to 𝑅²⁰, where 𝑅₀ denotes the circumradius of Γ_𝑁. Namely, 𝐿_𝑘=𝑅²⁰,𝑘=1,2,…,𝑁−1.(2.30)

Lemma 2.7. Suppose 𝐿_𝑘 is defined by Lemma 2.4, for any positive integer 𝑘,𝑗∶𝑘≥2,𝑘+𝑗≤𝑁−1, there exist constants 𝐶_{𝑘+𝑗,𝑗},𝐶_{𝑘−1,𝑗},𝐶_1,𝑗 which is only related to 𝑘,𝑗,𝑁 such that 𝐿_𝑘≤𝐶𝐶_{𝑘+𝑗,𝑗}𝐿_{𝑘+𝑗𝑘+𝑗,𝑗}+𝐶_{𝑘−1,𝑗}𝐿_𝑘−1+𝐶_1,𝑗𝐿₁,+𝐶_{𝑘−1,𝑗}+𝐶_1,𝑗=1.(2.31) A sufficient condition of equality is that Γ_𝑁 is a regular 𝑁-polygon in 𝑅³.

Proof. We prove it by mathematical induction with respect to 𝑗.
(i) When 𝑗=1, let 𝐶_𝑘+1,1=𝐶_𝑘−1,1=𝑆_𝑘−1𝑆_𝑘+1/(2𝑆^2𝑘)(>0),𝐶_1,1=(𝑆₁/𝑆_𝑘)²(>0) from Lemma 2.4, we have𝐿_𝑘≤𝐶_𝑘+1,1𝐿_𝑘+1+𝐶_𝑘−1,1𝐿_𝑘−1+𝐶_1,1𝐿₁.(2.32) Let Γ_𝑁 be a regular 𝑁-polygon. By Remark 2.6, we know 𝐿_𝑘=𝐿_𝑘+1=𝐿_𝑘−1=𝐿₁=𝑅²⁰>0; it follows from Lemma 2.4 that the equality of (2.32) holds, thus, 𝐶_𝑘+1,1+𝐶_𝑘−1,1+𝐶_1,1=1.
(ii) Assume that Lemma 2.7 holds for 𝑗=𝑛≥1. Now we want to prove that Lemma 2.7 holds for 𝑗=𝑛+1. By the hypothesis 𝑘,𝑛+1∶𝑘≥2,𝑘+𝑛+1≤𝑁−1, we know that 𝑘,𝑛∶𝑘≥2,𝑘+𝑛≤𝑁−2≤𝑁−1. Thus, from the induction hypothesis, there exist constants 𝐶_{𝑘+𝑛,𝑛},𝐶_{𝑘−1,𝑛},𝐶_1,𝑛 such that𝐿𝐶_{𝑘+𝑛,𝑛}+𝐶_{𝑘−1,𝑛}+𝐶_1,𝑛=1,(2.33)_𝑘≤𝐶_{𝑘+𝑛,𝑛}𝐿_𝑘+𝑛+𝐶_{𝑘−1,𝑛}𝐿_𝑘−1+𝐶_1,𝑛𝐿₁.(2.34) A sufficient condition of equality of (2.34) is that Γ_𝑁 is a regular polygon with 𝑁 sides. Since 𝑘+1,𝑛∶𝑘+1≥3≥2,(𝑘+1)+𝑛≤𝑁−1, by induction hypothesis, substitute 𝑘 by 𝑘+1 in (2.33) and (2.34), in other words, there exist constants 𝐶^{∗𝑘+1+𝑛,𝑛},𝐶^∗𝑘,𝑛,𝐶^∗1,𝑛 such that 𝐿𝐶^{∗𝑘+1+𝑛,𝑛}+𝐶^∗𝑘,𝑛+𝐶^∗1,𝑛=1,(2.35)_𝑘+1≤𝐶^{∗𝑘+1+𝑛,𝑛}𝐿_𝑘+1+𝑛+𝐶^∗𝑘,𝑛𝐿_𝑘+𝐶^∗1,𝑛𝐿₁.(2.36) Substituting (2.36) into (2.32), we get the following inequality: 𝐿_𝑘≤𝐶_𝑘+1,1𝐶^{∗𝑘+1+𝑛,𝑛}𝐿_𝑘+1+𝑛+𝐶^∗𝑘,𝑛𝐿_𝑘+𝐶^∗1,𝑛𝐿₁+𝐶^𝑘−1,1𝐿_𝑘−1+𝐶_1,1𝐿₁.(2.37) Notice 0<𝐶_𝑘+1,1<1,0<𝐶^∗𝑘,𝑛<1,1−𝐶_𝑘+1,1𝐶^∗𝑘,𝑛>0. Solving inequality (2.37) with respect to 𝐿_𝑘, we obtain 𝐿_𝑘≤𝐶^{∗∗𝑘+𝑛+1,𝑛+1}𝐿_𝑘+𝑛+1+𝐶^{∗∗𝑘−1,𝑛+1}𝐿_𝑘−1+𝐶^{∗∗1,𝑛+1}𝐿₁,(2.38) where 𝐶^{∗∗𝑘+𝑛+1,𝑛+1}=𝐶^𝑘+1,1𝐶^{∗𝑘+𝑛+1,𝑛}1−𝐶_𝑘+1,1𝐶^∗𝑘,𝑛𝐶>0,_{∗∗𝑘−1,𝑛+1}=𝐶^𝑘−1,11−𝐶_𝑘+1,1𝐶^∗𝑘,𝑛𝐶>0,_{∗∗1,𝑛+1}=𝐶^𝑘+1,1𝐶^∗1,𝑛+𝐶_1,11−𝐶_𝑘+1,1𝐶^∗𝑘,𝑛>0.(2.39) Let Γ_𝑁 be a regular 𝑁-polygon. From Remark 2.6, we know 𝐿_𝑘=𝐿_𝑘+𝑛+1=𝐿_𝑘−1=𝐿₁=𝑅²⁰>0. It follows from Lemma 2.4 and induction hypothesis that the equality of (2.38) holds. Thus, 𝐶^{∗∗𝑘+𝑛+1,𝑛+1}+𝐶^{∗∗𝑘−1,𝑛+1}+𝐶^{∗∗1,𝑛+1}=1, that is, Lemma 2.7 holds for 𝑗=𝑛+1. This completes the proof.

Lemma 2.8. Suppose 𝐿_𝑘 is defined by Lemma 2.4, we have the inequality as follows: 𝐿_𝑘≤𝐿₁,𝑘=2,3,…,𝑁−2.(2.40) A sufficient condition of equality is that Γ_𝑁 is a regular polygon with 𝑁 sides in 𝑅³.

Proof. Setting 𝑘+𝑗=𝑁−1 in (2.31), we get 𝐿_𝑘≤𝐶_{𝑁−1,𝑁−1−𝑘}𝐿_𝑁−1+𝐶_{𝑘−1,𝑁−1−𝑘}𝐿_𝑘−1+𝐶_{1,𝑁−1−𝑘}𝐿₁.(2.41) Since 𝐴_𝑖=𝐴_𝑗⇔𝑖≡𝑗(mod𝑁), 𝐿_𝑁−1∶=𝑁2𝑆_𝑁−1²^−1𝑁−1_𝑖=1||𝐴_𝑖𝐴_{𝑖+𝑁−1}||²=𝑁2𝑆₁²^−1𝑁−1_𝑖=1||𝐴_𝑖𝐴_𝑖−1||²=𝐿₁.(2.42) It follows from (2.41) and (2.42) that there exist constants 𝐶_𝑘−1,𝐶₁∶𝐶_𝑘−1+𝐶₁=1 such that 𝐿_𝑘≤𝐶_𝑘−1𝐿_𝑘−1+𝐶₁𝐿₁,(2.43) where 𝐶_𝑘−1=𝐶_{𝑘−1,𝑁−1−𝑘}>0,𝐶₁=𝐶_{𝑁−1,𝑁−1−𝑘}+𝐶_{1,𝑁−1−𝑘}>0, for any 𝑘∈{2,3,…,𝑁−2}. Using (2.43) repeatedly, we get 𝐿_𝑘≤𝐶_𝑘−1𝐿_𝑘−1+𝐶₁𝐿₁≤𝐶_𝑘−1𝐶^∗𝑘−2𝐿_𝑘−2+𝐶^∗1𝐿₁+𝐶₁𝐿₁=𝐶^{∗∗𝑘−2}𝐿_𝑘−2+𝐶^1∗∗𝐿₁≤𝐶^{∗∗∗𝑘−3}𝐿_𝑘−3+𝐶^1∗∗∗𝐿₁≤⋯≤𝐶𝐿₁,(2.44) therefore, 𝐿_𝑘≤𝐶𝐿₁.(2.45) Set Γ_𝑁 is a regular polygon with 𝑁 sides in 𝑅³. By Remark 2.6, we know 𝐿_𝑘=𝐿₁=𝑅²⁰>0; from Lemma 2.7, we have the equality of (2.45) holds, which implies that 𝐶=1. A sufficient condition of equality of (2.40) is that Γ_𝑁 is a regular 𝑁-polygon in 𝑅³. This completes the proof.

Lemma 2.9. Let Γ=Γ(𝐴𝐵)∶𝑟=𝑟(𝑡)=𝑥(𝑡)𝑖⃗+𝑦(𝑡)𝑗⃗+𝑧(𝑡)𝑘⃗(𝑎≤𝑡≤𝑏) be a smooth curve with the end points 𝐴 and 𝐵 in 𝑅³. If the function 𝑓∶Γ(𝐴𝐵)→𝑅 is Riemann integrable on Γ(𝐴𝐵), considering a partition of Γ(𝐴𝐵) by means of 𝑁−1(𝑁≥3) points 𝐴=𝐴₀,𝐴₁,…,𝐴_𝑖−1,𝐴_𝑖,…,𝐴_𝑁=𝐵 such that |𝐴₀𝐴₁|=|𝐴₁𝐴₂|=⋯=|𝐴_𝑖−1𝐴_𝑖|=⋯=|𝐴_𝑁−1𝐴_𝑁|=|Γ_𝑁|/𝑁, where Γ_𝑁∶𝐴₀𝐴₁⋯𝐴_𝑁−1𝐴_𝑁 is a broken line, we have 𝑀(𝑓,Γ(𝐴𝐵))∶=1||Γ(𝐴𝐵)||_Γ(𝐴𝐵)𝑓𝑑𝑠=lim_𝑁→∞1𝑁^𝑁−1_𝑖=0𝑓𝐴_𝑖.(2.46)

Proof. Since Γ(𝐴𝐵) is a smooth curve with the end points 𝐴 and 𝐵 in 𝑅³, ||Γ(𝐴𝐵)||=^𝑏𝑎𝑥^2(𝑡)+𝑦′²(𝑡)+𝑧^2(𝑡)𝑑𝑡(2.47) exists and |Γ(𝐴𝐵)|>0. And 𝑓∶Γ(𝐴𝐵)→𝑅 is Riemann integrable on Γ(𝐴𝐵), it follows that 𝑓∶Γ(𝐴𝐵)→𝑅 is bounded on Γ(𝐴𝐵), that is, there exists a constant 𝑀₁>0 such that |𝑓(𝑃)|≤𝑀₁ for any 𝑃∈Γ(𝐴𝐵). For any 𝑖∈{0,1,2,…,𝑁−1}, lim_𝑁→∞||Γ𝐴_𝑖𝐴_𝑖+1||=lim_𝑁→∞||𝐴_𝑖𝐴_𝑖+1||=lim_𝑁→∞||Γ_𝑁||𝑁=lim||Γ(𝐴𝐵_𝑁→∞)||𝑁=0.(2.48) It follows from the definition of the curve integral _Γ(𝐴𝐵)𝑓𝑑𝑠=lim_{𝑁→∞𝑁−1}_𝑖=0𝑓𝑃_𝑖||Γ𝐴_𝑖𝐴_𝑖+1||,∀𝑃_𝑖∈Γ𝐴_𝑖𝐴_𝑖+1(2.49) that _Γ(𝐴𝐵)𝑓𝑑𝑠=lim_{𝑁→∞𝑁−1}_𝑖=0𝑓𝐴_𝑖||Γ𝐴_𝑖𝐴_𝑖+1||.(2.50) Since lim_𝑁→∞(|Γ(𝐴_𝑖𝐴_𝑖+1)|/|𝐴_𝑖𝐴_𝑖+1|)=1(∀𝑖∈{0,1,2,…,𝑁−1}), for any 𝜀>0, there exists 𝑚∈𝑍(𝑚>1) such that 1≤|Γ(𝐴_𝑖𝐴_𝑖+1)|/|𝐴_𝑖𝐴_𝑖+1|<1+𝜀, for 𝑁>𝑚, therefore 0≤||Γ𝐴_𝑖𝐴_𝑖+1||−||𝐴_𝑖𝐴_𝑖+1||<𝜀||𝐴_𝑖𝐴_𝑖+1||=𝜀||Γ^𝑁||𝑁≤𝜀||Γ𝑁(𝐴𝐵)||.(2.51) Thus, |||||_𝑁−1_𝑖=0𝑓𝐴_𝑖||Γ𝐴_𝑖𝐴_𝑖+1||−^𝑁−1_𝑖=0𝑓𝐴_𝑖||𝐴_𝑖𝐴_𝑖+1|||||||=|||||^𝑁−1_𝑖=0≤_𝑁−1𝑓𝐴_𝑖||Γ𝐴_𝑖𝐴_𝑖+1||−||𝐴_𝑖𝐴_𝑖+1|||||||_𝑖=0≤_𝑁−1||𝑓𝐴_𝑖||⋅||Γ𝐴_𝑖𝐴_𝑖+1||−||𝐴_𝑖𝐴_𝑖+1||_𝑖=0𝑀₁⋅𝜀||Γ(𝐴𝐵)||=𝑀₁𝑁||Γ(𝐴𝐵)||𝜀.(2.52) It follows that lim_{𝑁→∞𝑁−1}_𝑖=0𝑓𝐴_𝑖||Γ𝐴_𝑖𝐴_𝑖+1||=lim_{𝑁→∞𝑁−1}_𝑖=0𝑓𝐴_𝑖||𝐴_𝑖𝐴_𝑖+1||.(2.53) In view of (2.50), we get 𝑀(𝑓,Γ(𝐴𝐵))∶=1||Γ(𝐴𝐵)||_Γ𝑓𝑑𝑠=1||Γ(𝐴𝐵)||lim_{𝑁→∞𝑁−1𝑖=0}=1𝑓𝐴_𝑖||Γ𝐴_𝑖𝐴_𝑖+1||||Γ(𝐴𝐵)||lim_{𝑁→∞𝑁−1𝑖=0}𝑓𝐴_𝑖||𝐴_𝑖𝐴_𝑖+1||=lim_𝑁→∞1||Γ(𝐴𝐵)||^𝑁−1_𝑖=0𝑓𝐴_𝑖||Γ^𝑁||=lim_𝑁→∞𝑁||Γ_𝑁||||Γ(𝐴𝐵)||⋅lim^𝑁→∞1𝑁^𝑁−1_𝑖=0𝑓𝐴_𝑖=lim_𝑁→∞1𝑁^𝑁−1_𝑖=0𝑓𝐴_𝑖.(2.54) This completes the proof of Lemma 2.9.

Lemma 2.10. Let Γ be a smooth closed curve in 𝑅³. If the points 𝐴,𝐵 move continuously on Γ and keep |Γ(𝐴𝐵)| invariable, where 0<𝑙∶=|Γ(𝐴𝐵)|<|Γ|/2, we obtain 𝑀^[2]||𝐴𝐵||,Γ∶=1||Γ||_Γ||𝐴𝐵||²⋅𝑑𝑠≤||Γ||𝜋sin||𝑙Γ𝜋||.(2.55) A sufficient condition of equality is that Γ is a circle in 𝑅³.

Proof. By the theory of real number, there exists an increasing sequence of positive integers {𝑘_𝑛} such that lim_𝑛→∞𝑘_𝑛10^𝑛=𝑙||Γ||.(2.56) Write 𝑘∶=𝑘_𝑛,𝑁∶=10^𝑛. Consider a partition of Γ by means of 𝑁(𝑁≥10) points 𝐴₁,𝐴₂,…,𝐴_𝑖−1,𝐴_𝑖,…,𝐴_𝑁 such that |𝐴₁𝐴₂|=|𝐴₂𝐴₃|=⋯=|𝐴_𝑁−1𝐴_𝑁|=|𝐴_𝑁𝐴₁|=|Γ_𝑁|/𝑁, where Γ_𝑁∶𝐴₁𝐴₂⋯𝐴_𝑁−1𝐴_𝑁𝐴₁ is a polygon with 𝑁 sides.
First, we give the following fact: if the point 𝐴 moves to some point 𝐴_𝑖 along Γ, the point 𝐵 moves to 𝐴^∗𝑖 along Γ, in other words, 𝐴=𝐴_𝑖∈Γ,𝐵=𝐴^∗𝑖∈Γ,|Γ(𝐴_𝑖𝐴^∗𝑖)|=𝑙, thenlim_𝑛→∞|||𝐴_∗𝑖𝐴_𝑖+𝑘|||=0.(2.57) In fact, from lim_𝑛→∞Γ_𝑁(𝐴_𝑖𝐴_𝑖+𝑘)⊂Γ,|Γ(𝐴_𝑖𝐴^∗𝑖)|=𝑙, we only need to prove lim_𝑛→∞||Γ_𝑁𝐴_𝑖𝐴_𝑖+𝑘||=𝑙.(2.58) Since lim_𝑛→∞||Γ_𝑁𝐴_𝑖𝐴_𝑖+𝑘||=lim_𝑛→∞𝑘||𝐴^𝑖𝐴_𝑖+1||=lim_𝑛→∞𝑘||Γ^𝑁||=lim_𝑛→∞𝑁||Γ_𝑁||lim_𝑛→∞𝑘𝑁=||Γ||||Γ𝑙||=𝑙,(2.59)(2.58) holds, it follows that (2.57) holds.
From (2.57), we know for any 𝜀>0, there exists 𝑚∈𝑍(𝑚>1), when 𝑁>𝑚, such that|||||||𝐴_𝑖𝐴^∗𝑖|||²−||𝐴_𝑖𝐴_𝑖+𝑘||²|||||1||||<𝜀,𝑁^𝑁_𝑖=1|||𝐴_𝑖𝐴^∗𝑖|||²−1𝑁^𝑁_𝑖=1||𝐴_𝑖𝐴_𝑖+𝑘||²|||||=1𝑁|||||^𝑁_𝑖=1|||𝐴^𝑖𝐴^∗𝑖|||²−||𝐴_𝑖𝐴_𝑖+𝑘||²≤1|||||𝑁^𝑁_𝑖=1|||||||𝐴_𝑖𝐴^∗𝑖|||²−||𝐴_𝑖𝐴_𝑖+𝑘||²||||<1𝑁^𝑁_𝑖=1𝜀=𝜀.(2.60) By Lemma 2.9, we get 1||Γ||_Γ||𝐴𝐵||²𝑑𝑠=1||Γ||_Γ||𝐴𝐵||²𝑑𝑠_𝐴=lim_𝑛→∞1𝑁^𝑁_𝑖=1|||𝐴_𝑖𝐴^∗𝑖|||²=lim_𝑛→∞1𝑁^𝑁_𝑖=1||𝐴_𝑖𝐴_𝑖+𝑘||².(2.61) It follows from inequality (2.40) that 1𝑁^𝑁_𝑖=1||𝐴_𝑖𝐴_𝑖+𝑘||²≤1𝑁ssiinn((𝑘𝜋𝜋//𝑁𝑁))^2𝑁_𝑖=1||𝐴_𝑖𝐴_𝑖+1||²=1𝑁ssiinn((𝑘𝜋𝜋//𝑁𝑁))^2𝑁_𝑖=1||Γ^𝑁||𝑁²=1𝑁ssiinn((𝑘𝜋𝜋//𝑁𝑁))²||Γ_𝑁||²𝑁=𝑁sinsin(𝑘(𝜋𝜋/𝑁/𝑁))²||Γ_𝑁||².(2.62) It follows from (2.61) and (2.62) that 1||Γ||_Γ||𝐴𝐵||²𝑑𝑠=lim_𝑛→∞1𝑁^𝑁_𝑖=1||𝐴_𝑖𝐴_𝑖+𝑘||²≤lim_𝑛→∞𝑁sinsin(𝑘(𝜋𝜋/𝑁/𝑁))²||Γ_𝑁||²=sinlim^𝑛→∞(𝑘𝜋/𝑁)lim_𝑛→∞𝑁sin(𝜋/𝑁)²lim_𝑛→∞||Γ_𝑁||²=sin𝑙𝜋/||Γ||𝜋²||Γ||²=||Γ||𝜋sin||𝑙Γ𝜋||².(2.63) Namely, (2.55) holds. From Lemma 2.8, a sufficient condition of equality of (2.55) is that Γ is a circle in 𝑅³. This completes the proof.

Lemma 2.11 (Cauchy inequality [1, page 6]). Let Γ be a smooth closed curve in 𝑅³. If the function 𝑓∶Γ→𝑅 and 𝑔∶Γ→𝑅 are smooth, we have the inequality as follows: ||||_Γ𝑓⋅𝑔𝑑𝑠||||≤_Γ𝑓²𝑑𝑠⋅_Γ𝑔²𝑑𝑠.(2.64)

Lemma 2.12. Assume Γ is a closed Jordan curve in 𝑅³ and 𝑂∈𝑅³, then 𝐷(Γ)⊂Ω[𝑂,Γ].

Proof. Let 𝐴,𝐵∈𝑅³,𝐴≠𝐵, 𝐴𝐵 denote the straight line through the points 𝐴 and 𝐵, and let 𝐴𝐵 denote the oriented line segment with the initial point 𝐴 and terminal point 𝐵.
For any 𝑃∈𝐷(Γ), if 𝑃=𝑂, then 𝑃∈Ω[𝑂,Γ]; if 𝑃≠𝑂, then 𝑂𝑃𝑃∈∩𝑂𝐷𝑃(∩Γ)𝐷=(Γ)⟹∃𝑖∶𝑖≡_{𝑖≡1(mod2),1≤𝑖≤𝑘−1}1𝑄(_𝑖m𝑄od_𝑖+12,),𝑄_𝑖1∈≤Γ𝑖,≤𝑖=𝑘−1,12,,…𝑃∈,𝑘,𝑘≥2,𝑄_𝑖𝑄_𝑖+1.(2.65) If 𝑂∈𝑄_𝑖𝑄_𝑖+1, then 𝑃∈𝑄_𝑖𝑄_𝑖+1=𝑄_𝑖𝑂∪𝑂𝑄_𝑖+1⟹⟹𝑃𝑃∈∈Ω𝑄_𝑖[𝑂𝑂,⊂Γ]Ω.[𝑂,Γ]∨𝑃∈𝑂𝑄_𝑖+1⊂Ω[𝑂,Γ](2.66) If 𝑂∉𝑄_𝑖𝑄_𝑖+1, without loss of generality, we set |𝑂𝑄_𝑖|≤|𝑂𝑄_𝑖+1|, then 𝑃∈𝑄_𝑖𝑄_𝑖+1⊂𝑂𝑄_𝑖+1⊂Ω[𝑂,Γ]⟹𝑃∈Ω[𝑂,Γ].(2.67) In all, for any 𝑃∈𝐷(Γ)⇒𝑃∈Ω[𝑂,Γ], hence, 𝐷(Γ)⊂Ω[𝑂,Γ]. This completes the proof.

Lemma 2.13. For 𝑆{𝑂,Γ,Γ_𝑁}, we have the following inequality: 𝑀^𝑁[−2](𝑟)∶=1𝑁^𝑁_𝑖=1𝑟^𝑖−2^−1/2≤12⎷1𝑁^𝑁_𝑖=1||𝐴_𝑖𝐴_𝑖+1||²⋅cot𝜋𝑁,(2.68) with equality if and only if Γ_𝑁 is a regular 𝑁-polygon in 𝑅³ and 𝑂 is its central.

Proof. By Definition 1.5 and Lemma 2.12, for any 𝑃∈𝑉(Γ_𝑁), we have Ω[𝑂,Γ_𝑁]=𝐷(Γ_𝑁)⊂Ω[𝑃,Γ_𝑁]. Thus, ^𝑁_𝑖=1𝑟_𝑖||𝐴_𝑖𝐴_𝑖+1||=2||Ω𝑂,Γ_𝑁||=2||𝐷Γ_𝑁||≤2||Ω𝑃,Γ_𝑁||.(2.69) Using Cauchy inequality (∑^𝑁𝑖=1𝑎_𝑖𝑏_𝑖)²≤(∑^𝑁𝑖=1𝑎^2𝑖)(∑^𝑁𝑖=1𝑏^2𝑖), (2.69) and (2.8), we get ||Γ_𝑁||²=^𝑁_𝑖=1||𝐴_𝑖𝐴_𝑖+1||²=⎛⎜⎜⎝^𝑁_𝑖=1𝑟_𝑖||𝐴_𝑖𝐴_𝑖+1||⋅||𝐴_𝑖𝐴_𝑖+1||𝑟_𝑖⎞⎟⎟⎠²≤^𝑁_𝑖=1𝑟_𝑖||𝐴_𝑖𝐴_𝑖+1||^𝑁_𝑖=1||𝐴_𝑖𝐴_𝑖+1||𝑟_𝑖≤^𝑁_𝑖=1𝑟_𝑖||𝐴_𝑖𝐴_𝑖+1||⎷^𝑁_𝑖=1||𝐴_𝑖𝐴_𝑖+1||²⎷_𝑁_𝑖=1𝑟^𝑖−2≤2||Ω𝑃,Γ_𝑁||⎷^𝑁_𝑖=1||𝐴_𝑖𝐴_𝑖+1||²⎷_𝑁_𝑖=1𝑟^𝑖−2≤||Γ^𝑁||²2𝑁cot𝑁𝜋⎷^𝑁_𝑖=1||𝐴_𝑖𝐴_𝑖+1||²⎷_𝑁_𝑖=1𝑟^𝑖−2=||Γ^𝑁||²2cot𝑁𝜋⎷1𝑁^𝑁_𝑖=1||𝐴_𝑖𝐴_𝑖+1||²𝑀^𝑁[−2](𝑟)⁻¹.(2.70) Therefore, (2.68) holds. The equality occurs if and only if Γ_𝑁 is a regular 𝑁-polygon in 𝑅³ and 𝑂 is its central. This completes the proof.

Lemma 2.14 (Power Mean Inequality, [1, 7, 10, 11]). Let Γ be a smooth closed curve in 𝑅³. If the function 𝑓∶Γ→]0,∞[ is smooth, we have the following inequality, for 𝑝,𝑞∈𝑅,𝑝<𝑞, 𝑀^[𝑝](𝑓,Γ)≤𝑀^[𝑞](𝑓,Γ).(2.71)

Lemma 2.15. For 𝑆{Γ,Γ_𝑁}, we have 1𝑁^𝑁_𝑖=1sin𝑙^𝑖𝜋||Γ||≤sin𝑁𝜋.(2.72) If 𝑁=3 or 𝑁≥4,0<𝑙_𝑖≤|Γ|/4,𝑖=1,2,…,𝑁, we get the inequality as follows: 1𝑁^𝑁_𝑖=1sin𝑙^𝑖𝜋||Γ||²≤sin𝜋𝑁².(2.73) The equalities hold in (2.72) and (2.73) if and only if 𝑙₁=𝑙₂=…=𝑙_𝑁=|Γ|/𝑁.

Proof. By Definition 1.4, we know ∑^𝑁𝑖=1𝑙_𝑖=|Γ|. Since sin𝑥 is the concave function on ]0,𝜋/2[, it follows from Jensen inequality [17, 18] and 0<𝑙_𝑖<|Γ|/2(𝑖=1,2,…,𝑁,𝑁≥3) that (2.72) holds. When 𝑁=3, 𝑙₁𝜋/|Γ|,𝑙₂𝜋/|Γ|,𝑙₃𝜋/|Γ| are three inner angles of a triangle. See [1, page 205], (2.73) holds. Since (sin𝑥)² is the concave function on ]0,𝜋/4[, from 𝑁≥4,0<𝑙_𝑖≤|Γ|/4,𝑖=1,2,…,𝑁, (2.73) holds. In view of the condition of equality of Jensen inequality, the equalities of (2.72) and (2.73) hold if and only if 𝑙₁=𝑙₂=⋯=𝑙_𝑁=|Γ|/𝑁. This completes the proof.

3. Proof of Theorem 1.9

By (2.8),(2.64),(2.55), and (2.72) in order, we get _Γ≤||Ω𝑃,Γ_𝑁||𝑑𝑠_Γ||Γ_𝑁||²=14𝑁cot𝑁𝜋𝑑𝑠4𝑁cot𝑁𝜋_Γ||Γ_𝑁||²𝑑𝑠=14𝑁cot𝑁𝜋_Γ^𝑁_𝑖=1=1||𝐴_𝑖𝐴_𝑖+1||²𝑑𝑠4𝑁cot𝑁𝜋_Γ^𝑁_𝑖=1=1||𝐴_𝑖𝐴_𝑖+1||²+2_{1≤𝑖<𝑗≤𝑁}||𝐴_𝑖𝐴_𝑖+1||||𝐴_𝑗𝐴_𝑗+1||𝑑𝑠4𝑁cot𝑁𝜋^𝑁_𝑖=1_Γ||𝐴_𝑖𝐴_𝑖+1||²𝑑𝑠+2_{1≤𝑖<𝑗≤𝑁}_Γ≤||Γ||||𝐴_𝑖𝐴_𝑖+1||||𝐴_𝑗𝐴_𝑗+1||𝑑𝑠4𝑁cot𝑁𝜋⎛⎜⎜⎝^𝑁_𝑖=1∮_Γ||𝐴_𝑖𝐴_𝑖+1||²𝑑𝑠||Γ||+2_{1≤𝑖<𝑗≤𝑁}⎷∮_Γ||𝐴_𝑖𝐴_𝑖+1||²𝑑𝑠||Γ||⎷∮_Γ||𝐴_𝑗𝐴_𝑗+1||²𝑑𝑠≤||Γ||||Γ||⎞⎟⎟⎠4𝑁cot𝑁𝜋^𝑁_𝑖=1||𝜋Γ||sin𝑙^𝑖𝜋||Γ||²+2_{1≤𝑖<𝑗≤𝑁}||𝜋Γ||sin𝑙^𝑖𝜋||Γ||||𝜋Γ||sin𝑙^𝑗𝜋=||Γ||₃||Γ||4𝜋²𝑁cot𝑁𝜋^𝑁_𝑖=1sin𝑙^𝑖𝜋||Γ||²=||Γ||³𝑁4𝜋²cot𝜋𝑁𝑁1^𝑁_𝑖=1sin𝑙^𝑖𝜋||Γ||²≤||Γ||³𝑁4𝜋²cot𝜋𝑁sin𝑁𝜋²=||Γ||³𝑁8𝜋²sin2𝜋𝑁,(3.1)

which implies that 𝑀||Ω𝑃,Γ_𝑁||,Γ∶=1||Γ||_Γ||Ω𝑃,Γ_𝑁||𝑑𝑠≤𝑁||Γ||²8𝜋²sin2𝜋_Γ𝑁,𝑑𝑠_Γ𝑁||𝑃𝑄||𝑑𝑠_𝑄=2_Γ||Ω𝑃,Γ_𝑁||𝑑𝑠≤𝑁||Γ||³4𝜋²sin2𝜋𝑁.(3.2) Hence, (1.3) and (1.4) are proved. By this proof, a sufficient condition of equalities is that Γ is a circle and Γ_𝑁 is always a regular polygon with 𝑁 sides. This completes the proof of Theorem 1.9.

Remark 3.1. We may extend the concepts of the bounded cone surface and the satellite system without any central to 𝑅^𝑚(𝑚≥2). And Theorem 1.9 also holds for the satellite system without any central in 𝑅^𝑚.

4. Proof of Theorem 1.10

Since power mean 𝑀^𝑁[𝑡](𝑟) is increasing with respect to 𝑡 and by (2.68), we have the following inequalities, for 𝑡≤−2,𝑀^𝑁[𝑡](𝑟)≤𝑀^𝑁[−2](𝑟)≤12⎷1𝑁^𝑁_𝑖=1||𝐴_𝑖𝐴_𝑖+1||²⋅cot𝜋𝑁.(4.1) Write 𝑓∶=(1/𝑁)∑^𝑁𝑖=1|𝐴_𝑖𝐴_𝑖+1|². By (4.1), (2.71), (2.72) and (2.73), we get _Γ𝑀^𝑁[𝑡](𝑟)𝑑𝑠≤12cot𝑁𝜋_Γ⎷1𝑁^𝑁_𝑖=1||𝐴_𝑖𝐴_𝑖+1||²𝑑𝑠=||Γ||2cot𝑁𝜋||Γ1||_Γ=||Γ||𝑓^1/2𝑑𝑠2cot𝑁𝜋𝑀^[1/2](𝑓,Γ)≤||Γ||2cot𝑁𝜋=||Γ||𝑀^[1](𝑓,Γ)2cot𝑁𝜋1||Γ||_Γ𝑓𝑑𝑠=||Γ||2cot𝑁𝜋⎷1||Γ||_Γ1𝑁^𝑁_𝑖=1=||Γ||||𝐴_𝑖𝐴_𝑖+1||²𝑑𝑠2cot𝑁𝜋⎷1𝑁^𝑁_𝑖=11||Γ||_Γ||𝐴_𝑖𝐴_𝑖+1||²𝑑𝑠≤||Γ||2cot𝑁𝜋⎷1𝑁^𝑁_𝑖=1||𝜋Γ||⋅sin𝑙^𝑖𝜋||Γ||²=||Γ||²2𝜋cot𝑁𝜋⎷1𝑁^𝑁_𝑖=1sin𝑙^𝑖𝜋||Γ||²≤||Γ||²2𝜋cot𝑁𝜋⋅sin𝑁𝜋=||2Γ𝜋||²cos𝑁𝜋.(4.2) Therefore, (1.5) and (1.6) are proved. From this proof, the equalities of (1.5) and (1.6) occur if and only if Γ is a circle, 𝑂 is the central of the circle, and Γ_𝑁 is a regular polygon with sides 𝑁. This completes the proof of Theorem 1.10.

5. Applications

In Theorem 1.9, if both Γ and Γ_𝑁 are simple closed curves in 𝑅³, and Ω[Γ_𝑁] denotes the minimal surface [4, 5, 19–21] bounded by Γ_𝑁, then for any 𝑃∈𝑉(Γ_𝑁), |Ω[Γ_𝑁]|≤|Ω[𝑃,Γ_𝑁]|, where |Ω[Γ_𝑁]| denotes the area of Ω[Γ_𝑁]. Thus, we get the theorem as follows.

Theorem 5.1. For 𝑆{Γ,Γ_𝑁}, if both the curve Γ and the polygon Γ_𝑁 are simple closed curves in 𝑅³, we have 𝑀||ΩΓ_𝑁||,Γ∶=1||Γ||_Γ||ΩΓ_𝑁||𝑑𝑠≤𝑁||Γ||²8𝜋²sin2𝜋𝑁,(5.1) where Ω[Γ_𝑁] is a minimal surface bounded by Γ_𝑁 and |Ω[Γ_𝑁]| denotes its area. A sufficient condition of equality in (5.1) is that Γ is a circle and Γ_𝑁 is always a regular 𝑁-polygon.

In Theorem 5.1, let 𝑉(Γ_𝑁)={𝐴₁,𝐴₂,…,𝐴_𝑁} and satisfy |Γ(𝐴₁𝐴₂)|=|Γ(𝐴₂𝐴₃)|=⋯=|Γ(𝐴_𝑁−1𝐴_𝑁)|=|Γ(𝐴_𝑁𝐴₁)|=(|Γ|)/𝑁. Setting 𝑁→∞, we get the theorem as follows.

Theorem 5.2. If the curve Γ is a smooth simple closed curve in 𝑅³, we have ||Ω[Γ]||≤||Γ||²4𝜋,(5.2) where |Ω[Γ]| denotes the area of the minimal surface Ω[Γ] bounded by Γ. A sufficient condition of equality is that Γ is a circle.

Remark 5.3. If Γ is a smooth Jordan closed curve in 𝑅³, then Ω[Γ]=𝐷(Γ), thus, (5.2) is the well-known isoperimetric inequality [16, 22–33].

Definition 5.4. If Γ is a smooth Jordan closed curve in 𝑅³, 𝑂 is a fixed point in the interior of Γ and the point 𝑃 moves continuously on Γ, then 𝑆{𝑂,Γ,𝑃} is called the satellite system of single point with a central.
For 𝑆{𝑂,Γ,𝑃}, write 𝜌_𝑃∶=|𝑂𝑃|. 𝑙_𝑃 denotes the tangent line of Γ at the point 𝑃, and 𝑟_𝑃∶=Dis(𝑂,𝑙_𝑃) denotes the distance from the point 𝑂 to the tangent line 𝑙_𝑃.

Theorem 5.5. For 𝑆{𝑂,Γ,𝑃}, we have _Γ𝜌_𝑃𝑑𝑠≥_Γ𝑟_𝑃𝑑𝑠≥2||𝐷(Γ)||,(5.3) that is, 𝑀𝜌_𝑃,Γ≥𝑀𝑟_𝑃,Γ≥2||𝐷(Γ)||||Γ||.(5.4) The first equalities in (5.3) and (5.4) hold if and only if Γ is a circle and the point 𝑂 is the central of the circle; the second equalities hold if and only if Ω[𝑂,Γ]=𝐷(Γ).

Proof. Insert the inscribed polygon Γ_𝑁∶𝐴₁𝐴₂⋯𝐴_𝑁𝐴₁(𝑁≥3) such that ||𝐴₁𝐴₂||=||𝐴₂𝐴₃||=⋯=||𝐴_𝑁−1𝐴_𝑁||=||𝐴_𝑁𝐴₁||=||Γ^𝑁||𝑁,(5.5) then the point 𝑂 is in the inner of Γ_𝑁 when 𝑁 is sufficiently large. In the following, we assume that 𝑂 is in the inner of Γ_𝑁.
According to Definition 1.8, we only need to prove (5.3). Since for any 𝑃∈Γ, 𝜌_𝑃≥𝑟_𝑃, the first inequality of (5.3) holds. Let 𝑟_𝑖∶=Dis(𝑂,𝐴_𝑖𝐴_𝑖+1) denote the distance from the point 𝑂 to the line 𝐴_𝑖𝐴_𝑖+1. We havelim_𝑁→∞𝑟_𝑖=𝑟_𝐴𝑖,𝑖=1,2,3,…,lim_𝑁→∞1𝑁^𝑁_𝑖=1𝑟_𝐴𝑖=lim_𝑁→∞1𝑁^𝑁_𝑖=1𝑟_𝑖.(5.6) By Lemma 2.12, we know 𝐷(Γ_𝑁)⊂Ω[𝑂,Γ_𝑁]. Thus, ||𝐷Γ_𝑁||≤||Ω𝑂,Γ_𝑁||=^𝑁_𝑖=1||Δ𝐴_𝑖𝑂𝐴_𝑖+1||.(5.7) Equalities of (5.7) hold if and only if Ω[𝑂,Γ_𝑁]=𝐷(Γ_𝑁). By (5.6),(5.7), and Lemma 2.9, we get ||𝐷(Γ)||=lim_𝑁→∞||𝐷Γ_𝑁||≤lim_{𝑁𝑁→∞}_𝑖=1||Δ𝐴_𝑖𝑂𝐴_𝑖+1||=12lim^{𝑁𝑁→∞}_𝑖=1=1𝑟_𝑖||𝐴_𝑖𝐴_𝑖+1||2lim^{𝑁𝑁→∞}_𝑖=1𝑟_𝑖||Γ_𝑁||𝑁=lim^𝑁→∞||Γ_𝑁||2lim^𝑁→∞1𝑁^𝑁_𝑖=1𝑟_𝑖=||Γ||2lim^𝑁→∞1𝑁^𝑁_𝑖=1𝑟_𝐴𝑖=||Γ||2||Γ1||_Γ𝑟_𝑃𝑑𝑠=12_Γ𝑟_𝑃𝑑𝑠.(5.8) Observe that (5.8) implies _Γ𝑟_𝑃𝑑𝑠≥2||𝐷(Γ)||.(5.9) The second inequality of (5.3) is proved.
By this proof, the first equalities of (5.3) and (5.4) hold if and only if Γ is a circle and the point 𝑂 is the central of the circle; the second equalities hold if and only if Ω[𝑂,Γ]=𝐷(Γ). This completes the proof.

Definition 5.6. Theorem 5.5 may be explained as follows: the point 𝑂 denotes the central of earth; 𝑃 is a satellite and Γ denotes the moving trajectory of satellites; 𝑀(𝑟_𝑃,Γ) is the function mean of 𝑟_𝑃 on Γ and 𝑀(𝜌_𝑃,Γ) is the function mean of 𝜌_𝑃 on Γ.

Definition 5.7. For the 𝑆{𝑂,Γ,𝑃₁,𝑃₂}, which denotes the satellite system of two points with a central, in other words, there are only two satellites on same trajectory, we will discuss in another paper.

Acknowledgments

This work is supported by the Natural Science Foundation of China (no. 10671136), the Project of Science and Technology Department of Sichuan Province (no. 2011JY0077), and the Natural Science Foundation of Chengdu University of China (no. 2010XJZ28).