# 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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### 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 vote
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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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### 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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### 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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### 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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### 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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### 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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### 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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### 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 ...
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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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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) $x <... 0 votes 0 answers 17 views ### Assembly index of a collection of objects is subadditive to the distinct assembly indices? In assembly theory an assembly index of an object is the shortest path length from the root to an object apply some choice of operation to 'assemble' the object. For example, starting from an alphabet ... • 1,687 0 votes 0 answers 14 views ### Limits of a progression Question: "Show that a progression$(a_n)_{n\ge1}$applies for the following implication $$a_n \rightarrow a ~\text{when}~ n \rightarrow \infty, \Rightarrow |a_n| \rightarrow |a| ~\text{when}~ n \... 2 votes 1 answer 84 views ### Proving complex inequality I am struggling to prove the following complex inequality: \left | z+4 \right |-2\leq \left | z+2 \right |, \text {where} \:z \in \mathbb{C} My thought process so far is to use the triangle ... • 61 0 votes 0 answers 21 views ### Prove the following triangle inequality [duplicate] Prove that:$$ \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 views ### 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: $$\|{\bf x} - {\bf a}\|_{2}^{2} - \|{\bf y} - {\bf a}... • 31 0 votes 1 answer 61 views ### Triangle inequality involving distance between point and set I am trying to prove that if (X,d) is a metric space and A a non-empty subset of X, then \forall x,y \in X, it is satisfied that d(x,A) \leq d(x,y)+d(y,A). Obviously, in the particular case ... • 359 0 votes 4 answers 88 views ### Linear Algebra, Triangle Inequality Justification So I get that the triangle inequality shows that |z_1 + z_2| \leq |z_1| + |z_2|.But I do not understand how,$$z_2 \overline{z_1} + z_1 \overline{z_2} = 2\Re(z_2 \overline{z_1}) \leq 2∣z_2 \overline{... • 1 0 votes 0 answers 42 views ###$t \geq 1$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 ... • 207 1 vote 0 answers 114 views ### 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 views ### 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,... • 121 3 votes 1 answer 73 views ### 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$...
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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 ...