Bulletin of the
Korean Mathematical Society BKMS

In this paper, the reverse Bonnesen style inequalities for convex domain in the Euclidean plane $\mathbb R^2$ are investigated. The Minkowski mixed convex set of two convex sets $K$ and $L$ is studied and some new geometric inequalities are obtained. From these inequalities obtained, some isoperimetric deficit upper limits, that is, the reverse Bonnesen style inequalities for convex domain $K$ are obtained. These isoperimetric deficit upper limits obtained are more fundamental than the known results of Bottema (\cite{bottema}) and Pleijel (\cite{pleijel}).

Keywords: convex domain, the Minkowski mixed area, the isoperimetric deficit upper limit, the Bonnesen style inequality, the reverse Bonnesen style inequality

MSC numbers: Primary 52A10; Secondary 52A22