In Constant Curvature Surface Convex Domain Inverse Bonnesen Type Inequality

The main topics of this paper are reverse Bonnesen style inequalities and Bonnesen style inequalities in a surface of constant curvature. Firstly, based on the containment measure of a domain to contain another domain and the con-tainment measure inequality established by J Z.Zhou etc., a group of important inequalities with respect to the radius of the maximum inscribed disc and the minimum circumscribed disc, the length and area of a domain in a surface of constant curvature are obtained. As direct use of this group of inequalities, we give reverse Bonnesen inequalities which characterize the measure of a convex domain deviation from a geodesic disc. Moreover, a group of inequalities with respect to the maximum and the minimum of the curvature radius, the length and area of a strongly convex domain in a surface of constant curvature are given. By this group of inequalities, we also obtain reverse Bonnesen inequalities which characterize the measure of a strongly convex domain deviation from a geodesic disc. Secondly, we obtain Bonnesen style inequalities of a convex domain in a surface of constant curvature which are generalizations of Bonnesen style inequal-ities on the plane via these two group of inequalities. Thirdly, the relation of the first eigenvalue of any domain on hyperbolic surface to certain isoperimetric con-stants is given by us with the containment measure inequality. Lastly, we give a generalization of some isoperimetric inequalities on the plane to3dimensional Euclidean space.