Triangle Inequality · aiwiki.page

The triangle inequality is a fundamental relation in geometry and mathematical analysis stating that the distance between two points is no greater than the distance obtained by passing through a third point. For a distance function $d$ , it takes the form $d(x,z)\leq d(x,y)+d(y,z)$ . Related formulations apply to absolute values and vector norms. Although originally associated with triangle side lengths, the inequality is also a defining property of abstract distances and norms. (math.ucla.edu)

Geometric statement and origins

For a nondegenerate Euclidean triangle with side lengths $a,b,c$ , the sum of any two sides is strictly greater than the third:

a<b+c,\qquad b<c+a,\qquad c<a+b.

The classical statement appears in Book I, Proposition 20 of Euclid’s Elements. Euclid proves it by extending a side, constructing an isosceles triangle, and using the relationship between larger angles and their opposite sides. Thus an apparently intuitive fact receives a deductive geometric proof. (web.calstatela.edu)

When arbitrary points are allowed, including collinear or coincident points, the appropriate statement uses “less than or equal to.” For Euclidean distance, equality in $d(A,C)\leq d(A,B)+d(B,C)$ occurs precisely when $B$ lies on the closed line segment joining $A$ and $C$ . The distinction between strict and non-strict versions separates genuine triangles from degenerate configurations. (maths.usyd.edu.au)

Absolute values and complex numbers

For real numbers $x,y$ , the inequality becomes

|x+y|\leq |x|+|y|.

An elementary proof follows from $-|x|\leq x\leq |x|$ and the corresponding bounds for $y$ . Adding them gives $-(|x|+|y|)\leq x+y\leq |x|+|y|$ . Equality holds exactly when $xy\geq0$ , so the numbers have compatible signs or at least one is zero. (ms.uky.edu)

The same formula holds for complex numbers, with modulus replacing real absolute value. Viewing complex numbers as vectors in the plane makes this the Euclidean vector inequality. For nonzero complex numbers, equality occurs when they have the same argument: their associated vectors point in the same direction. (sites.chemengr.ucsb.edu)

Normed vector spaces and proof

A norm measures the size of a vector in a vector space. One of the defining conditions of a normed vector space is

\|u+v\|\leq\|u\|+\|v\|.

Here the inequality is an axiom of the abstract structure. For a proposed formula to qualify as a norm, however, this condition must be verified rather than merely assumed. (blogs.ncl.ac.uk)

For a norm induced by an inner product, a proof uses the Cauchy–Schwarz inequality:

\begin{aligned} \|u+v\|^2 &=\|u\|^2+2\operatorname{Re}\langle u,v\rangle+\|v\|^2\\ &\leq \|u\|^2+2\|u\|\|v\|+\|v\|^2\\ &=(\|u\|+\|v\|)^2. \end{aligned}

Taking nonnegative square roots proves the result. This works for both real and complex inner-product spaces. (sites.chemengr.ucsb.edu)

In Euclidean space, equality requires one vector to be a nonnegative scalar multiple of the other, unless either is zero. Other norms can have different equality conditions. For example, under $\|(x_1,x_2)\|_1=|x_1|+|x_2|$ , the vectors $(1,0)$ and $(0,1)$ attain equality despite being linearly independent. (maths.usyd.edu.au)

Metric spaces and the reverse inequality

A metric space consists of a set equipped with a distance function satisfying nonnegativity, separation of distinct points, symmetry, and the triangle inequality. Every norm supplies a metric through

d(x,y)=\|x-y\|,

because $x-z=(x-y)+(y-z)$ . A general metric space need not possess vector addition, straight lines, or angles; its triangle inequality is a relation among distances alone. (math.ucla.edu)

A useful consequence is the reverse triangle inequality:

\big|\|u\|-\|v\|\big|\leq\|u-v\|.

Indeed, applying the ordinary inequality to $u=(u-v)+v$ gives one bound, and interchanging $u,v$ gives the other. The analogous metric statement is

|d(x,z)-d(y,z)|\leq d(x,y).

Consequently, the norm and distance from a fixed point have Lipschitz continuity with constant $1$ ; small changes in position cannot produce larger changes in these quantities. This conclusion follows directly from the displayed bounds. (blogs.ncl.ac.uk)

Generalizations and analytical uses

Repeated application, formalized by mathematical induction, gives the finite-sum inequality

\left\|\sum_{k=1}^{n}v_k\right\| \leq\sum_{k=1}^{n}\|v_k\|.

Likewise, the distance between the endpoints of a finite chain is bounded by the sum of its successive distances. These are direct extensions of the two-term statement. (blogs.ncl.ac.uk)

For $L^p$ spaces, the triangle inequality is known as Minkowski’s inequality. For $1\leq p<\infty$ ,

\|f\|_p=\left(\int |f|^p\,d\mu\right)^{1/p}, \qquad \|f+g\|_p\leq\|f\|_p+\|g\|_p.

The integral is taken over the underlying measure space. The result also holds for $p=\infty$ , using the essential supremum norm. For $1<p<\infty$ , a standard proof applies Hölder’s inequality. (pi.math.cornell.edu)

In convergence arguments, splitting an error through an intermediate approximation yields

d(x_n,x)\leq d(x_n,y_n)+d(y_n,x).

Thus two quantities tending to zero bound a third. Similar estimates underlie reasoning about limits and Cauchy sequences, allowing separate approximation errors to be controlled without knowing their directions or signs. (math.ucla.edu)