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In mathematics, the Hadwiger–Finsler inequality is a result on the geometry of triangles in the Euclidean plane. It states that if a triangle in the plane has side lengths a, b and c and area T, then

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

Related inequalities

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

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

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:[1]

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

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

History

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

References

Leonard Mihai Giugiuc, Dao Thanh Oai and Kadir Altintas, An inequality related to the lengths and area of a convex quadrilateral, International Journal of Geometry, Vol. 7 (2018), No. 1, pp. 81 - 86, [1]

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