Wirtinger Inequality-the Geometric Applications

Isoperimetric inequality, Minkowski inequality and Wirtinger inequality are im-portant inequalities in Mathematics, especially in geometric analysis. The Wirtinger inequality plays an important role in solving geometric problems. This paper ex-plores applications of the Wirtinger inequality in solving geometric inequalities. The method could be extended to high dimension and to obtain some interesting results. We use the Wirtinger inequality to give a simplified proof of Minkowski inequality for mixed area of convex sets Ki and Kj, Then we give another simplified proof of the classical Isoperimetric inequality. This article is divided into three chapters:In chapter 1, we introduce some preliminaries, In particular, we introduce the perimeter and area of a convex set via support function. the spherical harmonic function and the mixed volume are also introduced.In chapter 2, Via the Wirtinger inequality, we obtain the Minkowski mixed area inequality of convex sets Ki and Kj. The main results are:Theorem 2.2(Minkowski) Let Ki,Kj are the convex sets of the perimeter Li,Lj, and of the area of Ai,Aj. Let Aij be the mixed area of Ki and Kj. Then we have (Minkowski inequality) Aij2≥AiAj. with the equality hold if and only if Ki and Kj are homothetic.Theorem 2.5(Zhou) LetΓbe a simple curve of length L bounding a convex domain D of area A in Euclidean plane R2. If the curvatureκofΓis positiveThe equality holds if and only ifΓbounds a disc. Chapter 3 refers to the Wirtinger inequality on the sphere, We extend the Wirtinger Inequality to the sphere. The main results are:Theorem 3.3 Let Ki,Kj are the convex bodies of the curvature integral Mi,Mj, and of the surface area of Si, Sj. Let Sij be the mixed surface area of Ki and Kj. Then we have Sij2≥SiSj. with the equality hold if and only if Ki and Kj are homothetic.