Some Notes On The Planar Isperimetric Deficit

In this paper, we investigate the isoperimetric deficit of the convex body in Euclidean plane R2via the containment measure in integral geometry. We obtain some upper bounds and lower bounds (which are also called the Bonnesen-type inequalities) of the isoperimetric deficit by the maximum width, the minimum width and the mean width of the convex body. These upper bounds are more general than Bottema’s and Pleijel’s results.