Hadwiger–Finsler inequality
Encyclopedia
In mathematics
, the Hadwiger–Finsler inequality is a result on the geometry
of triangle
s in the Euclidean plane, named after the mathematician
s Hugo Hadwiger
and Paul Finsler
. It states that if a triangle in the plane has side lengths a, b and c and area A, then
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 A, then
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.
The Hadwiger–Finsler inequality is a special case of Pedoe's inequality
.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, the Hadwiger–Finsler inequality is a result on the geometry
Geometry
Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers ....
of triangle
Triangle
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 segments. A triangle with vertices A, B, and C is denoted ....
s in the Euclidean plane, named after the mathematician
Mathematician
A mathematician is a person whose primary area of study is the field of mathematics. Mathematicians are concerned with quantity, structure, space, and change....
s Hugo Hadwiger
Hugo Hadwiger
Hugo Hadwiger was a Swiss mathematician, known for his work in geometry, combinatorics, and cryptography.-Biography:...
and Paul Finsler
Paul Finsler
Paul Finsler was a German and Swiss mathematician.Finsler did his undergraduate studies at the Technische Hochschule Stuttgart, and his graduate studies at the University of Göttingen, where he received his Ph.D. in 1919 under the supervision of Constantin Carathéodory...
. It states that if a triangle in the plane has side lengths a, b and c and area A, then
Weitzenböck's inequality
Weitzenböck's inequality
In mathematics, Weitzenböck's inequality states that for a triangle of side lengths a, b, c, and area \Delta, the following inequality holds:Equality occurs if and only if the triangle is equilateral...
is a straightforward corollary
Corollary
A corollary is a statement that follows readily from a previous statement.In mathematics a corollary typically follows a theorem. The use of the term corollary, rather than proposition or theorem, is intrinsically subjective...
of the Hadwiger–Finsler inequality: if a triangle in the plane has side lengths a, b and c and area A, then
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
If and only if
In logic and related fields such as mathematics and philosophy, if and only if is a biconditional logical connective between statements....
the triangle is an equilateral triangle, i.e. a = b = c.
The Hadwiger–Finsler inequality is a special case of Pedoe's inequality
Pedoe's inequality
In geometry, Pedoe's inequality, named after Daniel Pedoe, states that if a, b, and c are the lengths of the sides of a triangle with area ƒ, and A, B, and C are the lengths of the sides of a triangle with area F, then...
.