Hadwiger-Finsler inequality

In mathematics, the Hadwiger-Finsler inequality is a result on the geometry of triangles in the Euclidean plane, named after the mathematicians Hugo Hadwiger and Paul Finsler. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and Geometry ( Greek γεωμετρία; geo = earth metria = measure is a part of Mathematics concerned with questions of size shape and relative position A triangle is one of the basic Shapes of Geometry: a Polygon with three corners or vertices and three sides or edges which are Line Euclidean geometry is a mathematical system attributed to the Greek Mathematician Euclid of Alexandria. A mathematician is a person whose primary area of study and research is the field of Mathematics. Hugo Hadwiger (1908 &ndash 1981 was a Swiss Mathematician. He is known for Hadwiger's theorem in Integral geometry, and a number of conjectures It states that if Δ is a triangle in the plane with side lengths a, b and c and area A, then

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

Weitzenböck's inequality is a straightforward corollary of the Hadwiger-Finsler inequality: if Δ is a triangle in the plane with side lengths a, b and c and area A, then

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

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 Δ is an equilateral triangle, i. In Mathematics, Weitzenböck's inequality states that for a triangle of side lengths a b c and area \Delta the following A corollary is a statement which follows readily from a previously proven statement In Geometry, Heron's (or Hero's formula states that the Area (A of a Triangle whose sides have lengths a, b, and ↔ Properties The area of an equilateral triangle with sides of length a\\! e. a = b = c.

External links

Hadwiger-Finsler inequality at PlanetMath. PlanetMath is a free, collaborative online Mathematics Encyclopedia.
Weizenbock's inequality at PlanetMath. PlanetMath is a free, collaborative online Mathematics Encyclopedia.

References

Finsler, Paul; Hadwiger, Hugo (1937). "Einege Relationen im Dreieck". Commentarii Mathematici Helvetici 10 (1): 316–326. doi:10.1007/BF01214300. A digital object identifier ( DOI) is a permanent identifier given to an Electronic document.

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