Hadwiger–Finsler inequality

• Weitzenböck's inequality is a straightforward corollary of the Hadwiger–Finsler inequality: if a triangle in the plane has side lengths a, b and c and area T, then

${\displaystyle a^{2}+b^{2}+c^{2}\geq 4{\sqrt {3}}T\quad {\mbox{(W)}}.}$

Hadwiger–Finsler inequality is actually equivalent to Weitzenböck's inequality. Applying (W) to the circummidarc triangle gives (HF)[1]

Weitzenböck's inequality can also be proved using Heron's formula, by which route it can be seen that equality holds in (W) if and only if the triangle is an equilateral triangle, i.e. a = b = c.

• A version for quadrilateral: Let ABCD be a convex quadrilateral with the lengths a, b, c, d and the area T then:[2]

${\displaystyle a^{2}+b^{2}+c^{2}+d^{2}\geq 4T+{\frac {{\sqrt {3}}-1}{\sqrt {3}}}\sum {(a-b)^{2}}}$ with equality only for a square.

Where ${\displaystyle \sum {(a-b)^{2}}=(a-b)^{2}+(a-c)^{2}+(a-d)^{2}+(b-c)^{2}+(b-d)^{2}+(c-d)^{2}}$

The Hadwiger–Finsler inequality is named after Paul Finsler and Hugo Hadwiger (1937), who also published in the same paper the Finsler–Hadwiger theorem on a square derived from two other squares that share a vertex.

• List of triangle inequalities
• Isoperimetric inequality

• Finsler, Paul; Hadwiger, Hugo (1937). "Einige Relationen im Dreieck". Commentarii Mathematici Helvetici. 10 (1): 316–326. doi:10.1007/BF01214300.CS1 maint: ref=harv (link)
• Claudi Alsina, Roger B. Nelsen: When Less is More: Visualizing Basic Inequalities. MAA, 2009, ISBN 9780883853429, pp. 84-86

• Proof of Hadwiger-Finsler inequality at PlanetMath.
• Weizenbock's inequality at PlanetMath.