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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.

MSC:

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
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