- Cezar Lupu, Cosmin Pohoata, Sharpening Hadwiger-Finsler’s inequality, Crux Mathematicorum with Mathematical Mayhem, 2 (2008), 97-101. (Link to the extracted PDF file); based on this article, in 2008, the authors were awarded with the Traian Lalescu Medal for their undergraduate research activity (though technically I am not an undergraduate yet).
The Hadwiger-Finsler inequality is known in the literature as a generalization of the following: In any triangle ABC with side lengths a, b, c and area S, we have that a^2+b^2+c^2 \geq 4S \sqrt{3}. This inequality is due to Weitzenbock (1991) [1], but also appeared in the International Mathematical Olympiad in 1961. Finsler and Hadwiger’s stronger version is as follows: a^2+b^2+c^2 \geq 4S \sqrt{3} + (a-b)^2+(b-c)^2+(c-a)^2.
In this note, we give a refinement of this inequality: In any triangle ABC with side lengths a, b, c, area S, inradius r, and circumradius R, the next inequality is valid: a^2+b^2+c^2 \geq 4S \sqrt{3+\frac{4(R-2r)}{4R+r}} + (a-b)^2+(b-c)^2+(c-a)^2.
We give a rather algebraic proof using an equivalent form of Schur’s inequality. In the last section, we give some basic applications of our result.
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