Some Bonnesen-style triangle inequalities. (English) Zbl 1033.52008

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.


52A40 Inequalities and extremum problems involving convexity in convex geometry
52A38 Length, area, volume and convex sets (aspects of convex geometry)
52B60 Isoperimetric problems for polytopes