A Strengthened Form Of Gage's Inequality And Its Analytic Version And The Stability Of Geometric Cauchy-Schwarz's Inequality

This thesis is composed of two parts. The first part concerns how to strengthen Gage's inequality as a more " isoperimetric-type" inequality, and the second part deals with the stability of geometric Cauchy-Schwarz's inequality.In order to study the popular curve shortening flow in the plane, Gage[3] has shown in 1983 the following inequality, if k is the signed curvature of a closed convex plane curve γ with length L and enclosing area A, then one getsGage calls it an isoperimetric inequality in [3], but he does not show that the equality holds if and only if the curve γ is a circle, while as an isoperimetric-type inequality, one should prove this kind of result. Here, we will try our best to use the unit-speed outward normal flow to prove this result, and therefore to strengthen Gage's inequality as a more " isoperimetric-type" inequality. It will be the main task of the first part of the present thesis. At the same time, we will use Minkowski's support function to restate Gage's inequality as an integral inequality which can be thought of as an analytic version of Gage's inequality.In the second part, we will first recall the notions of the stability of geometric inequalities, and then, deal with the stability problem of geometric Cauchy-Schwarz's inequality. As we known, the total curvature of a simple closed plane curve γ is ±2π, that is, ∫γ kds = ±2π, from Cauchy-Schwarz'sinequality, one gets thatWe will use Minkowski's support function to study the stability of the above...