Some Bonnesen-style triangle inequalities (Q1407461)

Bonnesen-style inequalities are isoperimetric inequalities which provide a lower bound with some geometric meaning of the isoperimetric deficit \(L^2-4 \pi A \) of planar sets with rectifiable boundary. The first inequalities of this type were obtained by Bonnesen during the 1920's. The author considers the isoperimetric deficit of triangles \(L^2-12 \sqrt{3} A\) and obtains lower estimates in terms of the inradius and the circumradius. These bounds sharpen -- in the particular case of triangles -- the bounds obtained by Bonnesen for planar convex sets.