# Questions tagged [triangle-inequality]

The triangle inequality tells us that given points $x,y,z$ in a Euclidean space (or more generally, in a metric space), $d(x,z)\leq d(x,y)+d(y,z)$ ($d(a,b)$ refers to the distance between points $a$ and $b$). Use this tag on questions that rely on the triangle inequality in either a geometric or a metric space.

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### Is there a proof for triangle inequality in $\mathbb{R}$ by contradiction/absurd?

I want to prove that given $a,b,c\in\mathbb{R}$ we have $|a+b|\leq|a|+|b|$ using an absurd and reaching a contradiction. So, I state, by absurd, that $|a+b|>|a|+|b|$, but I can't reach the ...
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### Is the proof $|d(x, A) - d(y, A) |\le d(x, y)$ in $(X, d)$ and $\emptyset \neq A \subset X$ and $x, y\in X$ logically perfect?

$(X, d)$ be a metric space and $\emptyset \neq A \subset X$ and $x, y\in X$. To show: $|d(x, A) - d(y, A)| \le d(x, y)$ My attempt: \begin{align} &d(x, A) = \inf \{ d(x, a) : a\in A \}\\ &d(x,...
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### Find the greatest values of |z| such that |z-4/z|=2. Solution without triangle inequality

I have solved the problem $|z-\frac{4}{z}|=2$ using the triangle inequality, as seen elsewhere on this site. However, I have also seen an alternative solution, where the user has simply squared the ...
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### Is the function $f(x) =(\sum_{i=1}^{2}{|x_i|^{1/2}})^2$ , where $x=(x_1, x_2)\in {\mathbb{R^2}}$ a norm on $\mathbb{R^2}$?

$X=\mathbb{R^2}$ and $f:\mathbb{R^2} \to {\mathbb{R}}$ defined by $f(x) =(\sum_{i=1}^{2}{|x_i|^{1/2}})^2$ , where $x=(x_1, x_2)\in {\mathbb{R^2}}$. Does the function $f$ define a norm on $\mathbb{R^2}$...
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### Is $d(x, y)={(\sum_{i=1}^{n}{|{x_i} -{y_i}|}^p})^{1/p}$ , $x, y\in\mathbb{R^n}$, $0<p<1$ metric on $\mathbb{R^n}$?

$X=\mathbb{R^n}$ Define , $d:X×X\rightarrow\mathbb{R}$ by $d(x, y)={(\sum_{i=1}^{n}{|{x_i} -{y_i}|}^p})^{1/p}$ , $x, y\in\mathbb{R^n}$, $0<p<1$ Question: Is $d$ a metric on $\mathbb{R^n}$? ...
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### Proving triangle inequality for general vectors.

If we assume Cauchy-swarz $|\vec{u}||\vec{v}|\ge |\vec{u}\cdot\vec{v}|$, is the following proof for triangle inequality correct? $$|\vec{u}+\vec{v}|^2 = |\vec{u}|^2+|\vec{v}|^2+2(\vec{u}\cdot\vec{v})$$...
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### Triangle Equality is Equal When... [duplicate]

I'm trying to prove that the triangle inequality is equal when the two vectors $a, b$ are linearly dependent, but I'm failing to do that. I have as follows, \begin{align*} |x + y | = |x| + |y| &\...
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### How to Prove $\lim_{(x, y) \rightarrow (0, 0)}\frac{x^3-y^3}{x^2+y^2}=0$ [duplicate]

I wanted to prove how $$\lim_{(x, y) \to (0, 0)} \frac{x^3-y^3}{x^2+y^2} = 0.$$ Specifically, I want to use the Squeeze Theorem for multivariable calculus. Then I know that I should pick an ...
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### Does the following inequality hold - the inner product divided by the product of norms?

Let $\cdot$ denotes the dot product and $||\boldsymbol{x}||$ denotes the $L^2-$norm of the vector $\boldsymbol{x}$. Suppose $\boldsymbol{a,b,c}$ are vectors in $\mathbb{R}^3$. Does the following ...
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### Doubt in a step involving triangle inequality

This is a step in page 323 of the book Convex Optimization by Stephen Boyd. I am trying to understand how we move from 2nd to 3rd step. So far, I understood that we have to use triangle inequality and ...
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### How to prove a Triangle Inequality

Let $f:\,\mathbb{R}^2\rightarrow\mathbb{R}_+$ defined via $f(x) = \frac{\|x\|^2}{w\cdot x}$, where $w\in\mathbb{R}^2$ is fixed and the dot denotes the inner product. I want to show (or disprove) that, ...
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### Is it possible to choose six triples of lengths from $1,2,3,\ldots, 20$ to form six triangles with equal perimeters?
Rachael has 20 thin rods whose lengths, in centimeters, are $1, 2, 3, \ldots, 20$. Any two rods can be connected at their ends. Rachael selects three rods to make a triangle, then three other rods to ...