Curvature Integral Inequalities For Oval And The Isoperimetric Deficit For Polygon

As a well known result, The isoperimetric theorem states that:for a simple closed curveΓ(in the euclidian plane) of length L enclosing a domain of area A, we have the inequality Equality is attained if and only if this curve is a euclidean circle. This means that among the set of domains of fixed area, the euclidean circle has the smallest perimeter or among the set of domains of fixed perimeter, the euclidean circle has the biggest area.In general, The isoperimetric inequality do not involve the curvature, Then, Gage gives integral inequality involving the square of the curvature for convex curve. ThatLetΓbe a oval curve of length L in the plane R2, and A the area enclosed byΓ. Then the curvatureκofΓsatisfy the following inequality: Equality holds if and only ifΓis a circle.The Ros isoperimetric theorem states in the plane that:LetΓbe a oval curve in the plane R2, and A the area enclosed byΓ. Then the curvatureκofΓsatisfy the following inequality: Equality holds if and only ifΓis a circle.In the first part of this paper, we investigate some flat oval curve of curvature integral inequalities. First, an integral inequality of function is derived, and on the basis of this integral inequality strengthen Ros Inequality form has been obtained;Theorem 3.2.1 Let f be is a C2-function of period 2πand f02πfdt=0, then Equality holds if and only if f(t)=a cos t+b sin t,where a, b are constants.Theorem 3.2.2 LetΓbe a oval curve of length L in R2, and A the area enclosed byΓ. Then the curvatureκofΓsatisfy the following inequality: Equality holds if and only ifΓis a circle.Next, by the properties of outside parallel convex set and monotonicity of the function a simple proof of Gage Isoperimetric Inequalities and Entropy Inequality for curvature are given, and thus we obtain a new integral on the curvature of the isoperimetric inequality;Theorem 3.4.3 LetΓbe a oval curve of length L in the plane R2, and A the area enclosed byΓ. Then the curvatureκofΓsatisfy the following inequality: Equality holds if and only ifΓis a circle.By Jensen inequality and the oval support functions we obtain a series of weak curvature integral inequalities;Theorem 3.5.2 LetΓbe a oval curve of length L in the plane R2, and f(x) is a strictly convex function on (0,+∞). Then the curvatureκofΓsatisfy the following inequality: Equality holds if and only ifΓis a circle.Continuously, we do some discussion about series of the curvature integral inequalities for oval in the paper.Theorem 3.6.1 LetΓbe a oval curve of length L in the plane R2. Then the curvatureκofΓsatisfy the following inequalities: Where Each equality sign holds if and only ifΓis a circle.Theorem 3.6.2 LetΓbe a oval curve of length L in the plane R2. Then the curvatureκof F satisfy the following inequalities: Where Each equality sign holds if and only ifΓis a circle.Theorem 3.6.3 LetΓbe a oval curve of length L in the plane R2, then the curvatureκofΓfor every positive integer 1≤l≤m≤n satisfy the following inequalities: Where Each equality sign holds if and only ifΓis a circle.In fact, There is a more basic problem for polygons than the classical isoperimetric problem.for polygons, There is also the corresponding discrete isoperimetric Inequality:(the isoperimetric inequality for polygons) LetΓn be an n-gon (a polygon with n sides) of perimeter Ln and area An, the following inequality is known Equality is attained if and only if the n-gon is regular.In geometry,△(Γ)= L2-4πA is called the isoperimetric deficit of the curveΓ. It measures the "deviation ofΓfrom circularity". Similarly, We can also define the isoperimetric deficit△(Γn)= L2-4cnA of the polygonΓn. It measures the "deviation ofΓn from the n-gon is regular".In the second part, we have a discussion about isoperimetric deficit for plane polygon. First of all, according to X. M. Zhang's idea two plane polygon's Bonnesen-type inequalities are obtained:Theorem 4.2.5 LetΓn be a convex polygon of length Ln in the plane R2, and An the area enclosed byΓn. Then the in-radius R ofΓn satisfy the following inequalities, there Each equality sign holds if and only ifΓn is a regular n-gon.Lastly, by geometrical relationship for the polygon some Bottema-type inequalities for planar polygons have been obtained:Theorem 4.3.4 LetΓn be a convex polygon of length Ln in R2, and An the area enclosed byΓn. Then in-radius ri and circum-radius re ofΓn satisfy the following inequalities, Each equality sign holds if and only ifΓn is a regular n-gon.