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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2answers
55 views

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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39 views

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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69 views

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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1answer
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Proving $|a| + |b| \leq |a+b| + |a-b|$ using the Triangle Inequality

Prove $$|a| + |b| \leq |a+b| + |a-b|$$ using the Triangle Inequality. Struggling a bit with this question, here's the little progress I made. $|a+b| \leq |a| + |b|$ $|a-b| \leq |a| + |b|$ $|a+b| + |a-...
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1answer
28 views

How to prove cosine distance does not satisfy triangle inequality metric in Euclidean space

I am trying yo prove cosine distance doesn't satisfy metric rule i know $$cosine \ distance = D(x,y)=1−S(x,y) = 1- \frac{x⋅y}{||x||||y||}$$ and able to proof cosine distance indeed satisfy other 3 ...
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4answers
131 views

How can we visualize $a^2+b^2-c^2$ for a triangle of sides $a$, $b$, $c$?

Let $a$, $b$, and $c$ be three lengths of sides of a triangle, that is, $a+b>c$. How can we visualize the value $a^2+b^2-c^2$ as length of some segment or area, ... constructed from the triangle $...
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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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2answers
130 views

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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2answers
67 views

Show the $d_p(x,y)$ is not a metric on $\mathbb{R}^2$

Question: For $0\lt p\lt 1$ and $x, y \in\mathbb{R}^2$ with $x=(\xi_1, \xi_2), y = (\eta_1, \eta_2)$, let $d :\mathbb{R}^2 \times \mathbb{R}^2 \rightarrow \mathbb{R} $ Be defined as $d_p(x, y) = (|\...
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1answer
34 views

Bounding an integral involving tail probabilities

I am currently reading through Van der Vaart and Wellner's book on empirical process theory. In chapter 2.9, they define the quantity $$||\xi||_{2,1} \equiv \int_0^\infty \sqrt{P(|\xi| > x)}\,\...
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1answer
66 views

When does the norm of the sum of $n$ vectors equal the sum of the norms of the vectors?

I'm trying to prove the following: Let $V$ be an inner product space over $\mathbb{R}$ or $\mathbb{C}$. For vectors $x_1,\ldots,x_n$, $$\left\lvert\sum\limits_{i=1}^n x_i\right\rvert = \sum\limits_{i=...
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1answer
58 views

Let $z_1,z_2$ be complex numbers with $|z_1|=|z_2|=1$. Prove that $|z_1 + 1| + |z_2 + 1| + |z_1z_2 + 1| \geq 2$ [duplicate]

Let $z_1,z_2$ be complex numbers with $|z_1|=|z_2|=1$. Prove that :- $$|z_1 + 1| + |z_2 + 1| + |z_1z_2 + 1| \geq 2$$ What I Tried:- From Triangle Inequality, we have :- $$|z_1 + 1| + |z_2 + 1| + |...
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0answers
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In a direction iterative method $x^{k+1}=x^k + d^k$ can one conclude that there is no limit point if $d^k$ is bounded away from zero?

When in an interactive direction method for solving a problem in $\mathbb{R}^n$ (be it optimization, zero of function, etc), the difference of the iterates is given by $$ x^{k+1} - x^k = t^kd^k $$ For ...
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1answer
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How does the triangle inequality prove continuous? [closed]

In my Complex Analysis textbook it states that a function $f$ of the complex argument $z$ is continuous iff it is continuous viewed as a function of the two real variables $x$ and $y$. Then it states ...
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1answer
69 views

Is there an inequality that involves $\|x+y\|$ and $\|x\|\|y\|$?

I was working on a problem where I had reached a step $$c^2\|x+y\|+ 2 \varepsilon \|T\|^2 \|x\|\|y\|$$ This made me curious and I just wanted to know if there were any inequalities in general that ...
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1answer
47 views

$2\sum_{i,j} |x_i-y_j| \geq \sum_{i,j} (|x_i-x_j| + |y_i-y_j|)$?

Let $x_1, \ldots, x_n$ and $y_1, \ldots, y_n$ be real numbers. Question: Is $$2\sum_{i,j=1}^n |x_i-y_j| \geq \sum_{i,j=1}^n (|x_i-x_j| + |y_i-y_j|)$$ true? Motivation: This is motivated from a ...
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1answer
29 views

Prove the metric of $\ell_2$ satisfies the triangle-inequality

Problem: Prove the metric of $\ell_2$ satisfies the triangle-inequality Reminders: A point in $ \ell_2 $ space is an infinite sequence $ \mathbf x = \langle x_k \rangle_{k=1}^{\infty} $ of real ...
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0answers
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Metric of a sequence (triangle inequality)

Define the function $d:X^w \times X^w \longrightarrow \mathbb{R}$ by \begin{equation*} d(x,y) = \begin{cases} 0 \text{ if } x = y \\ \frac{1}{n} \text{ if } x\not =y \text{ and } n \text{ is the ...
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3answers
102 views

Geometry Problem from Olympiad book [closed]

Given the base and the vertical angle of a triangle show that its area is greatest when it's isoceles I am stuck on how to proceed and which theorem to use here Thanks in advance
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1answer
34 views

Why does this inequality involving norms hold?

We are given $n$ linearly independent vectors $\lbrace \Phi_1, \dots, \Phi_n \rbrace$ with $||{\Phi_i}||=1$. We consider any linear combination $\sum_{i=1}^n \alpha_i \Phi_i$. My linear algebra script ...
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0answers
39 views

Is the triangle inequality preserved when taking the minimum of both sides?

Consider the statement $$\min\|a+b\|\leq \min\|a\|+\min\|b\|$$ where the minimum is taken over some set $C$ for which $a,b\in C$. Is this statement true? If not, what is a counterexample?
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Prove that if $a,b,c$ are sides of a triangle and $s$ is the semi-perimeter, then $a^2+b^2+c^2 \geq \dfrac{36}{35}\bigg(s^2 + \dfrac{abc}{s}\bigg)$

Prove that, if $a,b,c$ are the sides of a triangle and $s$ is the semi-perimeter, then $a^2+b^2+c^2 \geq \dfrac{36}{35}\bigg(s^2 + \dfrac{abc}{s}\bigg)$. What I Tried:- Nothing special really came in ...
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0answers
75 views

Parallelogram equality implies norm is induced by inner product

A useful property of a norm induced by an inner product is the parallelogram equality: $2\|x\|^2+2\|y\|^2=\|x+y\|^2+\|x-y\|^2$. This is trivial to show. It is also the case that if the parallelogram ...
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1answer
67 views

Show that $\rho$ satisfies triangle inequality.

Let $G \subseteq \mathbb C$ be an open set and $K \subseteq G$ be compact. Let $(\Omega,d)$ be a metric space and $C(G,\Omega)$ be the space of all continuous functions from $G$ to $\Omega.$ Define a ...
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1answer
43 views

Bounding the distance to the mean given the sum of squares distances

Let's say we have a set of points $P$, and let the mean of those points be $\bar{p}$, which is also the point that minimizes the sum of squared distances to the points of $P$. Let $opt$ denote the sum ...
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1answer
51 views

Prove or disprove $d(f,g):= \lVert f-g\rVert_p^p$ defines a metric on $L^p(\mu)$ for $0<p<1$. [duplicate]

Prove or disprove: $d(f,g):= \lVert f-g\rVert_p^p$ defines a metric on $L^p(\mu)$. My attempt: Use the fact that for $x,y\geq0$ and $0<p<1$, $$(x+y)^p\leq x^p+y^p$$ implies $$\int\vert{f+g}\...
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1answer
26 views

An absolute inequality

May I ask is the following correct? I say, $\forall c,a,\delta\in\mathbb{R},|c-a|-|\delta|\leq |c+\delta-a|\leq |c-a|+|\delta|$ is correct. Right side: $\begin{align*} |c+\delta-a|&=|c-a+\delta|\\ ...
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0answers
60 views

If the norm of the difference between two unit vector is lower bounded by a positive constant, does it mean that the inner product is upper bounded?

Let $x,y$ be two vectors with $\lVert x \rVert = \lVert y \rVert =1$ and $\lVert x-y \rVert \geq \delta$, where $\delta \gt 0$. Is it possible to show that, $1-(x^Ty)^2 \geq \delta^2$? My Approach: $$\...
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0answers
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Is this simple proof of the triangle inequality correct?

The triangle inequality states that for any complex numbers $z_1,z_2,\ldots,z_n$ we have $$\lvert z_1+z_2+\cdots+z_n\rvert\leqslant\lvert z_1\rvert+\lvert z_2\rvert+\cdots+\lvert z_n\rvert.$$ When I ...
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0answers
27 views

Reverse triangle inequality proof check

I have the following exercise: Show that if $||\cdot||$ is a norm on $X$ then $$\biggl|||x||-||y||\biggr|\leq ||x-y||$$ for all $x,y\in X$. My method of solving this: We have that $||x||=||x-y+y||\leq ...
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2answers
63 views

Uniform convergence and triangle inequality

Following @mathcounterexamples.net's answer on my previous question, I'm trying to rigorously prove $$\left\vert \frac{1}{n}\sum_{k=1}^{n}f(\frac{k}{n})-\frac{1}{n}\sum_{k=1}^{n}f_m(\frac{k}{n})\right\...
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1answer
63 views

Why is $ |1-a|+|a-(-1)| < 1+1$? [closed]

Why is $|1-a|+|a-(-1)| < 1+1$? This is for proving that $(-1)^n$ does not converge (for this prove you have to use the triangle inequality). You have to assume that $\lim\limits_{n->∞} (-1)^n = ...
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0answers
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How Does One Come Up With This Proof of The Triangle Inequality of Complex Integrals?

I saw this proof in the textbook and wonder what might have motivated it? Set $\theta = \arg \int^b_a f(t) \,dt$ then $$\left|\int^b_a f(t) \,dt\right|=e^{-i\theta}\int^b_a f(t) \,dt = \Re e^{-i\theta}...
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1answer
44 views

Triangle inequality for Bergman metric

Please help to prove triangle inequality for d(z,w) Where
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Proof by contradiction using triangle inequality | Sufficient to show non-integers?

I'm working on writing a proof for the solution, AB >= 3, to this problem: Given A,B,C are noncollinear points in a plane w/ integer coordinates such that AB, AC, and BC are integers, what is the ...
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0answers
8 views

Lipschitzness of the fractional powered euclidean norm .

Can someone help me to provide an upper or lower bound to the following? $$ \|\nabla_xf(x_1)\|^{\alpha}-\|\nabla_xf(x_2)\|^\alpha\leq?$$ where $0<\alpha<1$. I prefer a bound in terms of $\|\...
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2answers
122 views

Proving a norm of a set of continuous functions (Specifically the triangle inequality part)

I have the norm on the set of continuous functions from $s$ to $t$. Let $g$ and $h$ be in this set and define the dot product as $$\int_s^t g(x)h(x) \mathrm{dx}$$ and norm by $$ \|g\|_2 = \sqrt{g \...
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1answer
55 views

Find the minimum number of actions to make equal all the elements of the set.

Given a sorted set $S = \{x_1,x_2,...,x_n\}$, find the minimum number of actions to make equal all the elements of the set. Action stands for incrementing or decrementing some element of the set by $1$...
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1answer
15 views

How was the Triangle Inequality applied in this problem?

Let $(X, d)$ be a metric space and $E ⊂ X$ be a nonempty subset. Define a function $f : X → [0, ∞)$ by: $$f(x):= \inf \{d(x,y):y \in E \}$$ Prove that $f$ is uniformly continuous on $ X$ . Let me ...
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1answer
61 views

Find an interval for the side length of a triangle

The $\triangle ABC$ has $\left| AB \right| = \left| AD \right|$, $\angle BAD > 90^\circ$, $\left | AC \right | = 8$, $\left| DC \right| = 4$. Find an interval of possible values for $\left| BD \...
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1answer
35 views

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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0answers
21 views

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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1answer
62 views

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, ...
1
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1answer
104 views

Minkowski inequality for distances of random variables

I do not understand why Minkowski's inequality guarantee the triangle inequality for the next metric in a random variable set, with second moment finite: $d(X,Y)= \left \| X-Y \right \|_p$, where: $\...
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2answers
36 views

Proving triangle inequality involving measures and symmetric differences

I encountered the following exercise in a measure theory text: "Prove the triangle inequality $\mu(A \Delta C)\leq \mu(A \Delta B) + \mu(B \Delta C).$ Hint: Note that $1_{A\Delta B}=|1_{A}-1_{B}|....
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2answers
1k views

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 ...
4
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1answer
41 views

Does a continuous distance satisfy the triangle inequality?

I am not sure what tags are most appropriate here, so any help is appreciated (I found no tags for premetrics, quasimetrics, pseudometric, etc.). My level is undergrad-master. (TL;DR: One can probably ...