A Strengthened Reverse Isoperimetric Inequality And Its Stability Properties

Most geometric inequalities concerning convex bodies have the property that the occurence of the equality sign characterizes geometrically significant objects, like balls , ellipsoids . This fact suggests the following stability problem: If for some convex body the given inequality is satisfied so that it is not very different from an equality what can be said about the deviation of the body from these objects? Problems of this kind appear already in the work of Minkowski and Bonnesen, but have been investigated more systematically since the 1980s.This thesis deals with a strengthened reverse isoperimetric inequality and its stability properties, That is, for closed strictly convex plane curveγwith length p(γ) and area a{γ),where (a|~)(γ) denotes the oriented area of the domain enclosed by the locus of curvature centers ofγ, and the equality holds if and only ifγis a circle. Finally,we discuss the problem which is pointed out in article [18] using the method above mentioned.