In geometry , the triangle inequality is a theorem that says that one side of a triangle is at most as long as the sum of the other two sides. The “at most” includes the special case of equality. The triangle inequality also plays an important role in other areas of mathematics such as linear algebra or functional analysis .
Shapes of the triangle inequality
Triangle inequality for triangles
After the triangle is in the triangle , the sum of the lengths of two sides and always at least as great as the length of the third side . This means formally:
One can also say that the distance from A to B is always at most as large as the distance from A to C and from C to B combined, or to put it in a popular way: "The direct route is always the shortest."
The equals sign only applies if and sections are from - one also speaks of the triangle being "degenerate".
Since, for reasons of symmetry, the following also applies, it follows , analogously, that the total is
The left inequality is also sometimes referred to as the reverse triangle inequality .
The triangle inequality characterizes distance and magnitude functions . It is therefore used as an axiom of the abstract distance function in metric spaces .
Triangle inequality for real numbers
For real numbers the following applies:
- Because both sides of the inequality are not negative, squaring is an equivalence transformation :
- By deleting identical terms, we arrive at the equivalent inequality
- This inequality holds because for any
Inverse triangle inequality
As with the triangle, an inverse triangle inequality can be derived:
There is a substitution of there
if you set it instead, it results
so together (because for any real numbers and with and also applies )
If you replace with so you also get
Triangle inequality for complex numbers
The following applies to complex numbers :
- Since all sides are nonnegative, squaring is an equivalence transformation and one obtains
- where the overline means complex conjugation . If you delete identical terms and set so remains
- to show. With you get
- which is always fulfilled because of and the monotony of the (real) root function.
As in the real case, this inequality also follows
Triangle inequality of absolute value functions for bodies
Along with other demands, an amount function for a body is also given by the
established. It has to apply to all If all requirements (see article amount function ) are fulfilled, then an amount function is for the body
Is for all whole , then the amount is called non-Archimedean , otherwise Archimedean .
For non-Archimedean amounts, the
tightened triangle inequality
It makes the amount an ultrametric . Conversely, any ultrametric amount is non-Archimedean.
Triangle inequality for sums and integrals
Repeated application of the triangle inequality or complete induction results
for real or complex numbers . This inequality also applies when integrals are considered instead of sums:
If , where is an interval, Riemann integrable , then it holds
This also applies to complex-valued functions , cf. Then there is a complex number such that
is real, must be zero. Also applies
Triangle inequality for vectors
The following applies to vectors :
The validity of this relationship can be seen by squaring it
using the Cauchy-Schwarz inequality :
Here, too, it follows as in the real case
Triangle inequality for spherical triangles
The triangle inequality generally does not hold in spherical triangles .
However, it applies if you limit yourself to Eulerian triangles, i.e. those in which each side is shorter than half a great circle.
In the adjacent figure, the following applies
however is .
Triangle inequality for normalized spaces
In a normalized space , the triangle inequality becomes in the form
required as one of the properties that the standard must meet for everyone . In particular, it also follows here
for everyone .
In the special case of L p -spaces , the triangle inequality is called the Minkowski inequality and is proven by means of Hölder's inequality .
Triangle inequality for metric spaces
In a metric space , an axiom for the abstract distance function is required that the triangle inequality in the form
is fulfilled for all . In every metric space, the triangle inequality applies by definition. From this it can be deduced that in a metric space also the reverse triangle inequality
applies to all . In addition, the inequality
holds for any
↑ Harro Heuser: Textbook of Analysis, Part 1. 8th edition. BG Teubner, Stuttgart 1990, ISBN 3-519-12231-6 . Theorem 85.1
^ Walter Rudin: Real and Complex Analysis . MacGraw-Hill, 1986, ISBN 0-07-100276-6 . Theorem 1.33