Triangle inequality

A metric satisfies

d (x, z) \leq d (x, y) + d (y, z)

. A norm satisfies the corresponding inequality

∥ u + v ∥ \leq ∥ u ∥ + ∥ v ∥

.

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Reverse triangle inequality ( $∣ d (x, z) - d (y, z) ∣ \leq d (x, y)$ )

In a metric space, applying the triangle inequality in both directions gives

∣ d (x, z) - d (y, z) ∣ \leq d (x, y)

. In a normed vector space, its corresponding form is

∣∥ x ∥ - ∥ y ∥∣ \leq ∥ x - y ∥

. In particular, distance to a fixed point is a Lipschitz continuous real function.

Four-point triangle inequality

For four points

x_{1}, x_{2}, y_{1}, y_{2}

in a metric space,

d (x_{1}, y_{1}) - d (x_{1}, y_{2}) \leq d (y_{1}, y_{2}) \leq d (x_{2}, y_{1}) + d (x_{2}, y_{2}) .

Both steps follow from the triangle inequality. The first also holds with an absolute value around the difference, by the reverse triangle inequality. This transports a difference of distances measured from one point into a sum measured from another.

Triangle inequality for a vector-valued integral

For an integrable function

f

with values in a finite-dimensional normed vector space,

\int f \leq \int ∥ f ∥.

For the Euclidean norm, pair the integral with a unit vector in its direction and apply the Cauchy-Schwarz inequality pointwise.

Equality condition in the complex triangle inequality

For nonzero complex numbers

x, y

, equality

∣ x + y ∣ = ∣ x ∣ + ∣ y ∣

holds exactly when

x / y

is a positive real number, equivalently when the two numbers have the same complex argument modulo

2 π

.

Integral triangle inequality

For every integrable real or complex function,

\int f d μ \leq \int ∣ f ∣ d μ .

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Triangle inequality by

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The triangle inequality is a fundamental concept in geometry and mathematics that states the following for any triangle with sides of lengths \( a \), \( b \), and \( c \): 1. \( a + b > c \) 2. \( a + c > b \) 3. \( b + c > a \) In essence, the triangle inequality asserts that the sum of the lengths of any two sides of a triangle must be greater than the length of the remaining side.

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