On The Measures Of Asymmetry And Critical Points For Convex Bodies

The main topic is to deal with a sort of important affine invariants for convex bodies, including the dual p-measure of asymmetry for convex (star) bodies, the p-measures of asymmetry for convex bodies and the corresponding p-critical points.The concept of measures of asymmetry for convex bodies was introduced in1911, since then, many kinds of measure of asymmetry have been proposed and studied. Even recently some new measures of asymmetry were discovered. However, there are still many questions left.In this thesis, we carry out some fundamental researches on properties of p-critical points of convex bodies and the dual p-measure of asymmetry for convex (star) bodies. Our main work and results are as follows:First, we prove the continuity of p-critical points with respect to p on (1,+∞), discuss the connections between general p-critical points and the Minkowski-critical points (∞-critical points), and describe the behaviors of p-critical points of a sequence of convex bodies approximating to a convex bodies.Second, we define a family of measures of asymmetry for convex (star) body, called the dual p-measures of asymmetry, then reveal the connections between them and the p-dual mixed volumes, trying to find the best upper and lower bounds of the dual p-measures of asymmetry and the corresponding extreme bodies. At the end, we show some similar results to those for p-measures of asymmetry and give an example to show the difference between the dual p-measure and the p-measure of asymmetry.