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.
252
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How to prove the triangle inequality in the case of an n-sided polygon?
Basically, I want proof for
the statement that any particular side of an n-sided polygon is less than the sum of the lengths of its other sides. I tried proving it and I can successfully prove it for ...
1
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2
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Triangle Inequality for this metric
Let $(X,\varrho)$ be a metric space, and define $d:X\times X\to \mathbb{R}$ by
$$d(x,y)=\inf\lbrace\varrho(x,z)+\varrho(z,y):z\in X\rbrace$$
Show that $d$ is a metric on $X$.
I have all the axioms but ...
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Lemma for the Bescicovitch covering lemma
I want to prove the following lemma
Lemma (version 1) :
If $a,b \in \mathbb{R}^n$ are such that
$|a-b| > |a| > 0$
$|a - b| > |b| > 0$
then the angle between $a,b$ is bounded by below by a ...
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1
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How many triangles can be formed so that the area of $\triangle ABC$ is 9 times $\triangle DEF$? [closed]
I am currently working on an Olympiad math problem, and I am struggling to find a solution. I would greatly appreciate your help in solving this problem. I was unable to solve the problem because I ...
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Is there any simple way of proving triangle inequality for $d(x,y)= \dfrac{2|x-y|}{\sqrt{1+x^2}+\sqrt{1+y^2}}, \,\,x,y\in \mathbb{R}$ [duplicate]
$d(x,y)= \dfrac{2|x-y|}{\sqrt{1+x^2}+\sqrt{1+y^2}}, \,\,x,y\in \mathbb{R}$
Following are the approaches I took, but can't think further -
$d(x,y)= \dfrac{2|x-y|}{\sqrt{1+x^2}+\sqrt{1+y^2}}
= \dfrac{...
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1
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Proving that $\lVert x\rVert=\sqrt{\langle x,x \rangle}$ is a norm (triangle inequality).
To demonstrate that $\lVert x\rVert=\sqrt{\langle x,x \rangle}$ is a norm, I have to show that the properties of norms are coming from the properties of inner product. By example, let's demonstarte ...
- 177
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1
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Equality case in metric spaces
I have a question. Let $(X,d)$ a metric space. Then, for all $x,y \in X$, we have the triangle inequality : $d(x,y)+d(y,z)\geq d(x,z)$.
But, in what case we have the equality in the triangle ...
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L1 norm of (L1 normalized vector minus original vector) less than or equal to 1 minus the L1 norm of the original vector
I am reading one paper (https://arxiv.org/pdf/2208.09407.pdf). There is one step in the proof of Proposition B.1 that I can't understand. To simplify the notations, I restate the equation as follows ...
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Show that if $X_n \to X$ in $L^2$, $X_n$ is cauchy convergent in $L^2$.
This can be shown by using the Minkowski inequality:
$$
(E[|X_n - X_m|^2])^{1/2} \leqslant (E[(X_n - X)^2])^{1/2} + (E[(X_m - X)^2])^{1/2}
$$
Where both goes to zero by assumption and we have the ...
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How do I prove that $\left|b_n-b\right|<\frac{|b|}{2} $ implies $\left|b_n\right|>\frac{|b|}{2}$
It's a super important step in the Abbott Analysis book that:
$\left|b_n-b\right|<\frac{|b|}{2} $
implies
$\left|b_n\right|>\frac{|b|}{2}$
And I can't for the life of me figure how to prove it ...
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Interesting conundrum on the selecting elements from 1 to 1000 that can't form triangles
What is the largest number of distinct elements from set $\{1,2,3,4, \cdots, 1000\}$ that you can choose from such that no three of them are the side lengths of a triangle?
This was once in a high ...
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Show that $d(x, y) \ge 0$ whatever $x, y \in X$
Let $X$ be a set and $d : X \times X \to \mathbb R$ be a function satisfying the following conditions:
$d(x, x) = 0$ whatever $x \in X$;
$d(x, y) = d(y, x)$ whatever $x, y \in X$;
$d(x, z) \le d(x, y) ...
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1
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How to prove $|\sin x_1-\sin x_2|\leqslant |\tan x_1-\tan x_2|$ is correct?
I really have no clue how to solve this problem, can anyone help please?
Prove the equation:
$$|\sin x_1-\sin x_2| \leqslant|\tan x_1-\tan x_2|$$
I tried to use this rule:
If a function is continuous ...
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1
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1
answer
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Can someone prove to me if this inequality with norms and dot products holds?
Can someone show me if the following inequality holds ($\cdot$ is multiplication)?
$\|a\|\cdot\|b\| \le \frac{1}{2}(\|a\|^2 + \|b\|^2)$
I am sure the following holds ($\bullet$ is dot product):
$a\...
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Mahalanobis norm for multivariate normal distributions
For a multivariate normal $p(x,y)\sim N
\left(\begin{array}{c}
\mu_{x}\\
\mu_{y}
\end{array},\begin{array}{cc}
\Sigma_{xx} & \Sigma_{xy}\\
\Sigma_{yx} & \Sigma_{yy}
\end{array}\right)
$, and ...
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3
answers
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Range of $|z_1+z_2|$ for some $z_1,z_2\in\Bbb C$
So if we have 2 complex numbers $z_1$ and $z_2$, then the following inequality holds :
$$||z_1|-|z_2||\leq|z_1+z_2|\leq|z_1|+|z_2|$$
We did a question in class that went "Find the range of $|z|$ ...
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If $z_n\to x, $ why is $\lim_{n\to \infty}(d(x, z_n) +d(z_n, y)) = d(x, y)$ and not $\geq d(x, y)? $
In Probability Theory by Klenke, the author proves the following lemma:
Let $f, g: \Omega\to E$ be a measurable function with respect to $\mathcal A-\mathcal B(E). $ Then the map $H: \Omega\to [0,\...
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1
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1
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Proof of the equation for a rectangle in the Cartesian coordinate system with distances and absolute values [closed]
Can anyone help me find an approach to deriving the equation
$$
\left\lvert \frac{x}{p}+\frac{y}{q} \right\rvert + \left\lvert \frac{x}{p}-\frac{y}{q} \right\rvert = c
$$
for a rectangle in the ...
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1
answer
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Inequality with the sides of a triangle. [closed]
Let $a,b,c$ the sides of a triangle such that $a+b+c= \dfrac{1}{34^4}$. Show that:
$ \sqrt[5]{a+b-c} + 16\sqrt[5]{a+c-b} + \sqrt[5]{b+c-a} \leq 1$.
I put $a = x+y, b=y+z$ and $c=z+x$ with $(x,y,z) \in ...
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0
answers
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Deforming the "cross product metric"
It is well known that on $\mathbb{R}P^2$ one can define the following ''cross product metric''
$$d([x],[y]) = \frac{\|x \times y\|}{\|x\| \|y\|},$$
where $x,y \in \mathbb{R}^3 \setminus \{0\}$ and $\|....
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Would this be a valid proof for triangle inequality?
I am a first year Mathematics student. I have this homework where I have to prove the reverse triangle inequality $(||x|-|y|| ≤ |x-y| )$. I have proven it with the triangle inequality, so I had to ...
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Given a criteria function. Why two integral of cumulative function $\integ{F_1-F_2$ can be combine and the magic of triangle inequality
I have been studied a research for many days, but I have been confused by mathematical inequalities. I list the calculations as below and explain the notations.
\begin{equation}
\begin{split}
\left|{\...
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1
answer
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Proving that $||v||=\sqrt{2v_1^2+3v_2^2}$ defines a norm on $\mathbb{R}^2$
To prove a norm, I have to show 3 things:
for all $v,w\in V$ and $c\in \mathbb{R}$
(1) $||v||\ge 0$ along with $||v||=0$ iff $v=\textbf{0}$
(2) $||cv||=|c|*||v||$
(3) $||v+w||\le ||v||+||w||$
I have ...
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0
answers
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What is the length of $AF?$ [closed]
What is the length of $AF$?
Extend the side $BC$ of a parallelogram $ABCD$ to $E$ so that $C$ lies between
$B$ and $E$. Let the line segment $AE$ cut $BD$ and $CD$ at $F$, $G$ respectively.
Let $EG = ...
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How can I solve this complex inequality?
I'm starting to study complex numbers right now, and then I came across with this problem:
$$\frac{|z^m|}{|z^{n} +1|} < \frac{|R^m|}{|R^{n}-1|}$$
where $|z|=R$
I know that everything starts with ...
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2
answers
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Does there exists a continuous non-decreasing function $f$ such that $|x-z|\leq f(|x-y|)+|y-z|$?
Does there exist a continuous non decreasing function $f:[0,\infty)\longrightarrow \mathbb R$ with $g(0))=0$ such that
$$|x-z|\leq f(|x-y|)+|y-z|$$
for all $x,y,z\in X$ where $X$ is some subset of $\...
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0
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"Transform" of a semi-metric
Let $X$ be a set,
$d \, : \, X^2 \to [0,\infty) $ such that
$d(x,y) = 0 \iff x = y$
Let
$\hat{d}(x,y) := {\inf\{ \sum_{i = 1}^{n}{d(x_{i-1},x_i)} \; : \; x_0,x_1,\dots,x_n \in X \; , \; x_0 = x \, , \,...
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Is $ n \sum \rho_i^2 \geq (\sum \rho_i)^2$?
Say I have a real-valued vector $\rho$ of length $n$. Is the following inequality always true?
$n \sum \rho_i^2 \geq (\sum \rho_i)^2$
I originally thought this was a trivial triangle inequality like ...
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0
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Prove triangle inequality for inner product $\langle a,c\rangle \leq \langle a,b\rangle + \langle b,c\rangle $
For all $a,b,c$ which are vectors, determine whether $\langle a,c\rangle \leq \langle a,b\rangle + \langle b,c\rangle $ is true or false.
I have considered Cauchy-Schwartz inequality,
but there is a $...
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1
answer
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Prove that $|f'(z)| \leq \frac{1}{1-|z|}$
The problem is stated as:
Suppose $f: D[0,1] \mapsto D[0,1]$ is holomorphic. Then, prove for $|z| \leq 1$ the following: $$|f'(z)| \leq \frac{1}{1-|z|}$$
My attempt:
So, we can make use of Cauchy's ...
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1
answer
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Prove that: $\sum_{\text {sym}} a^3+3\sum_{\text{sym}} a^2b\ge 18$
Let $a,b,c$ be the sides of triangle, such that $abc=1$, then prove that:
$$\sum_{\text{sym}} a^3+3\sum_{\text{sym}} a^2b\ge 18.$$
Find at least one case where equality is possible.
My attempts:
I ...
- 622
1
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2
answers
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Prove $f: X \to \mathbb{R}_{\geq 0}$ where $f(x) = d(x, x_o)$ is a continuous function
I've read here: Proving that a sphere is a closed set that the function $f: X \to \mathbb{R}_{\geq 0}$ where $f(x) = d(x, x_o)$ is a continuous function and that it can be proven continuous via the ...
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1
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Triangle Inequality Holds IFF
I've just started learning about the triangle inequality. I've got the following two statements which at first glance look very similar.
(1) $x < y$ and $|x − y| ≤ | x − z| + | z − y|$
(2) $x <...
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Assembly index of a collection of objects is subadditive to the distinct assembly indices?
In assembly theory an assembly index of an object is the shortest path length from the root to an object apply some choice of operation to 'assemble' the object. For example, starting from an alphabet ...
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Limits of a progression
Question:
"Show that a progression $(a_n)_{n\ge1}$ applies for the following implication
$$a_n \rightarrow a ~\text{when}~ n \rightarrow \infty, \Rightarrow |a_n| \rightarrow |a| ~\text{when}~ n \...
2
votes
1
answer
84
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Proving complex inequality
I am struggling to prove the following complex inequality:
$\left | z+4 \right |-2\leq \left | z+2 \right |, \text {where} \:z \in \mathbb{C} $
My thought process so far is to use the triangle ...
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0
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21
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Prove the following triangle inequality [duplicate]
Prove that:
$$
\cos \left ( \sin x \right ) > \sin \left ( \cos x \right )
$$
First,we can prove that:
$$
\forall x\in \left ( 0 ,\frac{\pi }{2} \right ) ,\cos \left ( \sin x \right ) > \sin \...
0
votes
0
answers
67
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The upper bound of difference between two L2-norm squares
Suppose that ${\bf x}$, ${\bf y}$ are two vector variables, and ${\bf a}$ is a vector scalar, all of the same length. I wish to find $C$ such that:
$$\|{\bf x} - {\bf a}\|_{2}^{2} - \|{\bf y} - {\bf a}...
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votes
1
answer
61
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Triangle inequality involving distance between point and set
I am trying to prove that if $(X,d)$ is a metric space and $A$ a non-empty subset of $X$, then $\forall x,y \in X$, it is satisfied that $d(x,A) \leq d(x,y)+d(y,A)$. Obviously, in the particular case ...
- 359
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4
answers
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Linear Algebra, Triangle Inequality Justification
So I get that the triangle inequality shows that $|z_1 + z_2| \leq |z_1| + |z_2|$.But I do not understand how,
$$z_2 \overline{z_1} + z_1 \overline{z_2} = 2\Re(z_2 \overline{z_1}) \leq 2∣z_2 \overline{...
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0
answers
42
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$t \geq 1$ and triangle inequality
Let a, b and c be three distinct points in a vector space V with an inner product. If $d(a,c) = d(a,b) + d(b,c)$ then $c-a = t(b-a)$ with $t \geq 1$.
I'm trying to show (without success so far) that ...
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0
answers
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Another proof of Euler's inequality via the half-angle formulas
The Euler's inequality is an immediate consequence of Euler's identity in a triangle,
$$OI^2=R^2−2Rr.$$
An additional proof of the Euler's inequality is given at Elias Lampakis, Am Math Monthly, 122 ...
4
votes
2
answers
154
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Find minimum of $\sqrt{a^2+b^2}+\sqrt{b^2+c^2}+\sqrt{c^2+a^2}+ab+bc+ca$
Let $a, b,c$ be the lengths of the sides of a triangle such that $a+b+c=2$, find the minimum value of $$\sqrt{a^2+b^2}+\sqrt{b^2+c^2}+\sqrt{c^2+a^2}+ab+bc+ca$$
I don't have many ideas for this problem,...
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votes
1
answer
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Does triangle inequality imply Euclidean metric?
We know that the distances between vertices in a Euclidean space satisfy the triangle inequality, but is the converse true? Specifically, given a complete graph $K_n$ of $n>2$ vertices, along with $...
- 73
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0
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Scalar "Triangle Inequality" for Logarithms
I'm trying to create an algorithm that merges two sorted arrays of integers of lengths $m$ and $n$ with an overall time complexity of $T(n) = O\left(\log_2(n + m)\right)$, where $T(n)$ represents the ...
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0
answers
59
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Elementary proof of a generalized Triangle Inequality
My teacher gave this lemma in class without proof, I tried to prove it but found it seemed a little bit tricky:
Let a, b, c be real numbers. Then for any
$ \varepsilon\in(0, 1)$ we have $|\left| a-c\...
- 23
12
votes
1
answer
3k
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The theoretical importance of the half-angle formulas
Unlike the laws of sines, cosines and tangents, which are very well known, the half-angle formulas seem (although they appear timidly in the mathematical literature) not to enjoy the same popularity. ...
2
votes
2
answers
38
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Where in this statement is the reverse triangle rule used?
I've attached part of a proof from my lecture notes (the proof is showing that $\frac{1}{x^2}$ is continuous over $\mathbb{R} \setminus \{0\}$), it makes reference to using the reverse triangle rule ...
user999236
0
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0
answers
6
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Why does Minkowski's inequality acheive equality IFF added functions are positively linearly dependent?
Wikipedia stats on the Minkowski inequality:
Let $S$ be a measure space, let $1 \leq p < \infty$ and let $f$ and $g$ be elements of $L^p(S)$. Then $f + g$ is in $L^p(S)$, and we have the triangle ...
- 1,687
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0
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Mathematical expectation problem
Given: X is a random variable, $E|X|^\alpha<\infty$ with some $\alpha>0$ Prove that $E|X+b|^\alpha < \infty$ for any real b. Prove that $E|X+b|^\alpha = \infty$ if b-> $\infty$
My ...
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