Research On The Geometric Inequalities And Their Stabilities In The Plane

Let K be a compact convex set in the Euclidean space Rn.It is said to be a convex body if it has interior points.In this paper,we mainly study the strengthened forms of the Ros inequality and the stabilities of the generalized mixed width inequality for planar convex body,and the stability of the generalized dual mixed radial inequality for planar star body.The details are as follows:On one hand,Professor Zhou Jiazu proved the Ros inequality:let ? be a simple smooth closed curve with the length L in R2,and let K be a closed region with the boundary ? and the area A.If the curvature ? of ? is not zero everywhere,then(?)where s is arc length of ?,the equality holds if and only if ? is a circle.In the third section of the third chapter,by considering several classes of curves related to the convex curve ?,such as evolute curve,pedal curve and Wigner caustic,we use Fourier series to study the strengthened forms of the Ros inequality for the planar convex body.On the other hand,suppose that K and T are two planar convex bodies.Let A(K)and A(T)be the areas of K and T,respectively.Let ?k(K,?)and ?k(T,?)be their k-order width functions,respectively.Mao-Yang proved the generalized mixed width inequality:#12 the equality holds if and only if K and T are discs.In the second section of the fourth chapter,We obtain the stabilities of the generalized mixed width inequality by using L2-metric and Hausdorff distance.In addition,we also discuss the stability of the generalized dual mixed radial inequality for two planar star bodies.Suppose that P and Q are two planar star bodies.Let A(P)and A(Q)be the areas of P and Q,respectively.Let ?k(P,?)and ?k(Q,?)be their k-order radial functions,respectively.Mao-Yang proved the generalized dual mixed radial inequality.In the second section of the fifth chapter,We obtain the stability of the generalized dual mixed radial inequality by using dual L2-metric.