These inequalities for reduce to the following
:
:
The left-hand side polynomials of these two inequalities have roots that are the tribonacci constant In mathematics, the Fibonacci numbers form a sequence defined recursively by:
:F_n =
\begin
0 & n = 0 \\
1 & n = 1 \\
F_ + F_ & n > 1
\end
That is, after two starting values, each number is the sum of the two preceding numbers.
The Fibonacci seque ...
and its reciprocal. Consequently, is limited to the range where is the tribonacci constant.
Relationship with shortest paths
This generalization can be used to prove that the shortest curve between two points in Euclidean geometry is a straight line.
No polygonal path between two points is shorter than the line between them. This implies that no curve can have an arc length
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less than the distance between its endpoints. By definition, the arc length of a curve is the least upper bound of the lengths of all polygonal approximations of the curve. The result for polygonal paths shows that the straight line between the endpoints is the shortest of all the polygonal approximations. Because the arc length of the curve is greater than or equal to the length of every polygonal approximation, the curve itself cannot be shorter than the straight line path.[ p. 95.]
Converse
The converse of the triangle inequality theorem is also true: if three real numbers are such that each is less than the sum of the others, then there exists a triangle with these numbers as its side lengths and with positive area; and if one number equals the sum of the other two, there exists a degenerate triangle (that is, with zero area) with these numbers as its side lengths.
In either case, if the side lengths are ''a, b, c'' we can attempt to place a triangle in the Euclidean plane as shown in the diagram. We need to prove that there exists a real number ''h'' consistent with the values ''a, b,'' and ''c'', in which case this triangle exists.
By the Pythagorean theorem
In mathematics, the Pythagorean theorem or Pythagoras' theorem is a fundamental relation in Euclidean geometry between the three sides of a right triangle. It states that the area of the square whose side is the hypotenuse (the side opposit ...
we have and according to the figure at the right. Subtracting these yields . This equation allows us to express in terms of the sides of the triangle:
:
For the height of the triangle we have that . By replacing with the formula given above, we have
:
For a real number ''h'' to satisfy this, must be non-negative:
:
:
:
:
:
which holds if the triangle inequality is satisfied for all sides. Therefore there does exist a real number ''h'' consistent with the sides ''a, b, c'', and the triangle exists. If each triangle inequality holds strictly, ''h'' > 0 and the triangle is non-degenerate (has positive area); but if one of the inequalities holds with equality, so ''h'' = 0, the triangle is degenerate.
Generalization to higher dimensions
The area of a triangular face of a tetrahedron is less than or equal to the sum of the areas of the other three triangular faces. More generally, in Euclidean space the hypervolume of an - facet of an -simplex
In geometry, a simplex (plural: simplexes or simplices) is a generalization of the notion of a triangle or tetrahedron to arbitrary dimensions. The simplex is so-named because it represents the simplest possible polytope in any given dimension ...
is less than or equal to the sum of the hypervolumes of the other facets.
Much as the triangle inequality generalizes to a polygon inequality, the inequality for a simplex of any dimension generalizes to a polytope of any dimension: the hypervolume of any facet of a polytope is less than or equal to the sum of the hypervolumes of the remaining facets.
In some cases the tetrahedral inequality is stronger than several applications of the triangle inequality. For example, the triangle inequality appears to allow the possibility of four points , , , and in Euclidean space such that distances
:
and
:.
However, points with such distances cannot exist: the area of the 26–26–26 equilateral triangle is , which is larger than three times , the area of a 26–14–14 isosceles triangle (all by Heron's formula), and so the arrangement is forbidden by the tetrahedral inequality.
Normed vector space
In a normed vector space
In mathematics, a normed vector space or normed space is a vector space over the real or complex numbers, on which a norm is defined. A norm is the formalization and the generalization to real vector spaces of the intuitive notion of "leng ...
, one of the defining properties of the norm is the triangle inequality:
:
that is, the norm of the sum of two vectors is at most as large as the sum of the norms of the two vectors. This is also referred to as subadditivity. For any proposed function to behave as a norm, it must satisfy this requirement.[
]
If the normed space is euclidean, or, more generally, strictly convex, then if and
only if the triangle formed by , , and , is degenerate, that is,
and are on the same ray, i.e., or , or
for some . This property characterizes strictly convex normed spaces such as
the spaces with . However, there are normed spaces in which this is
not true. For instance, consider the plane with the norm (the Manhattan distance) and
denote and . Then the triangle formed by
, , and , is non-degenerate but
:
Example norms
*''Absolute value as norm for the real line
In elementary mathematics, a number line is a picture of a graduated straight line that serves as visual representation of the real numbers. Every point of a number line is assumed to correspond to a real number, and every real number to a po ...
.'' To be a norm, the triangle inequality requires that the absolute value satisfy for any real numbers and : which it does.
Proof:
:
:
After adding,
:
Use the fact that
(with ''b'' replaced by ''x''+''y'' and ''a'' by ), we have
:
The triangle inequality is useful in mathematical analysis
Analysis is the branch of mathematics dealing with continuous functions, limit (mathematics), limits, and related theories, such as Derivative, differentiation, Integral, integration, measure (mathematics), measure, infinite sequences, series (m ...
for determining the best upper estimate on the size of the sum of two numbers, in terms of the sizes of the individual numbers.
There is also a lower estimate, which can be found using the ''reverse triangle inequality'' which states that for any real numbers and :
:
*''Inner product as norm in an inner product space.'' If the norm arises from an inner product (as is the case for Euclidean spaces), then the triangle inequality follows from the Cauchy–Schwarz inequality
The Cauchy–Schwarz inequality (also called Cauchy–Bunyakovsky–Schwarz inequality) is considered one of the most important and widely used inequalities in mathematics.
The inequality for sums was published by . The corresponding inequality f ...
as follows: Given vectors and , and denoting the inner product as :
:
The Cauchy–Schwarz inequality turns into an equality if and only if and
are linearly dependent. The inequality
turns into an equality for linearly dependent and
if and only if one of the vectors or is a ''nonnegative'' scalar of the other.
:Taking the square root of the final result gives the triangle inequality.
* -norm: a commonly used norm is the ''p''-norm: where the are the components of vector . For the -norm becomes the ''Euclidean norm'': which is Pythagoras' theorem in -dimensions, a very special case corresponding to an inner product norm. Except for the case , the -norm is ''not'' an inner product norm, because it does not satisfy the parallelogram law. The triangle inequality for general values of is called Minkowski's inequality.[ ] It takes the form:
Metric space
In a metric space
In mathematics, a metric space is a set together with a notion of '' distance'' between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most general sett ...
with metric , the triangle inequality is a requirement upon distance:
:
for all , , in . That is, the distance from to is at most as large as the sum of the distance from to and the distance from to .
The triangle inequality is responsible for most of the interesting structure on a metric space, namely, convergence. This is because the remaining requirements for a metric are rather simplistic in comparison. For example, the fact that any convergent sequence in a metric space is a Cauchy sequence
In mathematics, a Cauchy sequence (; ), named after Augustin-Louis Cauchy, is a sequence whose elements become arbitrarily close to each other as the sequence progresses. More precisely, given any small positive distance, all but a finite numbe ...
is a direct consequence of the triangle inequality, because if we choose any and such that and , where is given and arbitrary (as in the definition of a limit in a metric space), then by the triangle inequality, , so that the sequence is a Cauchy sequence, by definition.
This version of the triangle inequality reduces to the one stated above in case of normed vector spaces where a metric is induced via , with being the vector pointing from point to .
Reverse triangle inequality
The reverse triangle inequality is an elementary consequence of the triangle inequality that gives lower bounds instead of upper bounds. For plane geometry, the statement is:[
]
:''Any side of a triangle is greater than or equal to the difference between the other two sides''.
In the case of a normed vector space, the statement is:
:
or for metric spaces, .
This implies that the norm as well as the distance function are Lipschitz continuous with Lipschitz constant , and therefore are in particular uniformly continuous.
The proof for the reverse triangle uses the regular triangle inequality, and :
:
:
Combining these two statements gives:
:
Triangle inequality for cosine similarity
By applying the cosine function to the triangle inequality and reverse triangle inequality for arc lengths and employing the angle addition and subtraction formulas for cosines, it follows immediately that
and
With these formulas, one needs to compute a square root for each triple of vectors that is examined rather than for each pair of vectors examined, and could be a performance improvement when the number of triples examined is less than the number of pairs examined.
Reversal in Minkowski space
The Minkowski space
In mathematical physics, Minkowski space (or Minkowski spacetime) () is a combination of three-dimensional Euclidean space and time into a four-dimensional manifold where the spacetime interval between any two events is independent of the ...
metric is not positive-definite, which means that can have either sign or vanish, even if the vector ''x'' is non-zero. Moreover, if ''x'' and ''y'' are both timelike vectors lying in the future light cone, the triangle inequality is reversed:
:
A physical example of this inequality is the twin paradox in special relativity
In physics, the special theory of relativity, or special relativity for short, is a scientific theory regarding the relationship between space and time. In Albert Einstein's original treatment, the theory is based on two postulates:
# The law ...
. The same reversed form of the inequality holds if both vectors lie in the past light cone, and if one or both are null vectors. The result holds in ''n'' + 1 dimensions for any ''n'' ≥ 1. If the plane defined by ''x'' and ''y'' is spacelike (and therefore a Euclidean subspace) then the usual triangle inequality holds.
See also
* Subadditivity
* Minkowski inequality
* Ptolemy's inequality
Notes
References
* .
* .
External links
{{DEFAULTSORT:Triangle Inequality
Geometric inequalities
Linear algebra
Metric geometry
Articles containing proofs
Theorems in geometry