The Lower Bounds of the Mixed Isoperimetric Deficit

Abstract

Let $$K_i$$ be plane convex bodies with the perimeters $$L_{K_i}$$ and areas $$A_{K_i}$$ for $$i=1,2$$ , respectively. In this paper, the lower bounds of the mixed isoperimetric deficit $$\Delta _{K_1,K_2}=L_{K_1}^2L_{K_2}^2-16\pi ^2 A_{K_1}A_{K_2}$$ are obtained; these bounds involve the areas enclosed by the evolute and the Wigner caustic, Hausdorff and $$L_2$$ distances between two convex bodies.