Skip to main content

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.

Filter by
Sorted by
Tagged with
0 votes
0 answers
17 views

How to prove that the convergence to a steady state exists if the change in transition probabilities are sufficiently slow?

Let's say we have a transition matrix $Q_{n}$ for each time step $n$ of a discrete Markov process, where it doesn't stay stationary for all time steps. I want to prove that if the change between each ...
magg13__'s user avatar
0 votes
0 answers
25 views

Help with my proof of the triangle inequality

I am trying to prove the triangle inequality for $\mathbb{R}^{n}$. I know there are proofs out there, but I would like help with my own proof. I have: For $ x,y \in \mathbb{R}^{n} $ $|x+y|^2 = \sum_{i=...
STRICKLAND_7's user avatar
1 vote
1 answer
53 views

How can we be sure that the triangle inequality works in this case?

Consider two complex numbers $z$ and $w$ such that $|z| = |w| = 3$ and $|z - w| = 3\sqrt{2}.$ What is the minimum value of $P = |z + 1 + i| + |w - 2 + 5i|$? This is a question from our country's ...
ten_to_tenth's user avatar
  • 1,078
0 votes
0 answers
14 views

Triangle inequality on a direct sum of Lebesgue spaces

I think a priori it is known that for $1 \leq p, q < \infty$ we have that $||(\hspace{0.1cm},)||: L^p(E) \times L^q(E) \rightarrow \mathbb{R}_{\geq 0} $, defined as $||(f, g)|| = (||f||^2_{L^p} + |...
bellumthirio's user avatar
1 vote
0 answers
30 views

Constructing such a $0<\delta<1$ that $|z + w|\leq \delta |z|+|w|$ when $0<\varepsilon\leq |\mathrm{Arg}(z)-\mathrm{Arg}(w)|\leq 2\pi -\varepsilon$

Let $z,w\in\mathbb{C}$ be two non-zero complex numbers such that for some $\varepsilon > 0$ we have $$0<\varepsilon\leq |\mathrm{Arg}(z)-\mathrm{Arg}(w)|\leq 2\pi -\varepsilon$$ I am trying to ...
Cartesian Bear's user avatar
2 votes
1 answer
79 views

Conjecture: The line joining the incenter and the circumcenter always subtends an obtuse angle at centroid

Consider the triangle formed by joining the circumcenter, the incenter and the centroid of a triangle (is there is already a name for this triangle in literature?). Simulations show that the line ...
Nilotpal Sinha's user avatar
2 votes
1 answer
76 views

How do you prove $\lvert x - y \rvert < 1$ then $\lvert x\rvert<\lvert y\rvert +1 $?

How do you prove $\lvert x - y \rvert < 1$ then $\lvert x\rvert<\lvert y\rvert +1 $? I know this proof has the form of the triangle inequality, but I can't seem to figure it out. This is from ...
nnabahi's user avatar
  • 27
1 vote
0 answers
67 views

Is this a metric space?

Let $d: \mathbb{C}\times\mathbb{C} \to \mathbb{C}$ defined by $d(x,y) = \frac{|x-y|}{\sqrt{1+|x|^2}+\sqrt{1+|y|^2}}$ All the properties of $d$ can easily be satisfied. I proved that $d$ is a bounded ...
Hamza Ayub's user avatar
4 votes
0 answers
124 views

Necessary and sufficient conditions for a product of norms to be the square of a norm

Let $(E,N)$ be a normed space, and $M$ be another norm. Can you find a necessary and sufficient condition on $M$ so that $\sqrt{NM}$ is also a norm on $E$? I hink the condition is that $M$ is ...
J.Mayol's user avatar
  • 790
1 vote
2 answers
67 views

inequality $|x + 3\eta^2| \geq |x| + |\eta|^2$

I'm learning the basics of Fourier Analysis. Reading a book, I've found this inequality: " If $x\geq -3$ and $|\eta | >2$ we have $|x + 3\eta^2| \geq |x| + |\eta|^2$. I've tried to solve this ...
Nick's user avatar
  • 97
-2 votes
1 answer
26 views

General Triangular Inequality for distance between subsets. [closed]

Suppose we have $A$ and $B$ as subsets of a metric space $(E, d)$. Is it true that for any subset $C$ of $E$, $d(A, B) \leq d(A, C) + d(C, B)$?
anton's user avatar
  • 9
2 votes
1 answer
65 views

Prove the inequality for x,y real numbers

I'm trying to prove this inequality using the triangle inequality, but unfortunately haven't had much luck. I feel like I can rewrite the left hand side into something usable, but I don't know what to....
rlaivsezlt's user avatar
0 votes
0 answers
160 views

Problems with proving exponential distance function

I want to show when ${G(A,B)=\frac{1}{m}\sum_{i=1}^m}\frac{1}{\frac{1}{n}\sum_{j=1}^{n}e^{-\vert{a_i-b_j}\vert}}$, ${G(A,C)\leq G(A,B) \cdot G(B,C)}$. I think from $\vert a_i-b_k+b_k-c_j \vert \leq \...
Jmp3r's user avatar
  • 11
2 votes
1 answer
105 views

Triangle Inequality for Le Cam's Metric?

Let $P, Q$ be probability distributions (say absolutely continuous with respect to the Lebesgue measure for simplicity). The Le Cam metric (also sometimes called the Triangular Discrimination) is ...
Mark Schultz-Wu's user avatar
-1 votes
2 answers
56 views

convergence vs cauchy sequence

I was looking at a proof that every sequence in $\mathbb{R}$ converges if and only if it is a Cauchy sequence. It starts like this: Suppose a sequence ($a_n$) converges to $A$, fix $\epsilon > 0$, ...
Eugenio Laguna's user avatar
0 votes
1 answer
31 views

Estimation theorem/triangle inequality

How do I use the reverse triangle inequality for this function? |z|=R $$|z^4 +5z^2+4|$$ From constructing a triangle, I obtain that $$|z^4|< |z^4 +5z^2+4| + |5z^2| +|4|$$ And thus $$|z^4 +5z^2+4| &...
jensen paull's user avatar
0 votes
1 answer
50 views

How can I continue reducing this absolute value inequality's denominator

I need to prove the following inequality: $|\sqrt{|a|+1}-\sqrt{|b|+1}|\le \dfrac{|a-b|}{2}$ I started from the LHS and multiplied it by $\sqrt{|a|+1}+\sqrt{|b|+1}$ and obtained the following result: $\...
Fairuz_'s user avatar
  • 129
1 vote
1 answer
91 views

The triangle inequality of the $L^p$-norm for a infinite sum

I just wonder that $\|\sum _{i=1}^\infty f_i\|_p\leq\sum _{i=1}^\infty\|f_i\|_p$. Where $\|.\|_p$ is the $L^p$- norm ,that is, $\|f\|_p=(\int_U|f|^pdm)^{1/p}$ Does this always work, or under what ...
topst's user avatar
  • 147
0 votes
0 answers
48 views

Prove that sum of any two sides of a triangle is greater than twice the median with respect to the third side.

"Prove that sum of any two sides of a triangle is greater than twice the median with respect to the third side." This question is from Mathematics NCERT Exemplar Class 9, exercise 7.4 , ...
Gaurav Lodhi's user avatar
0 votes
0 answers
63 views

Assume that ($a_n$) converges to a and ($b_n$) converges to b. Prove that ($a_n-b_n$) converges to $a-b$.

I used a similar mean to the one that solves $a_n+b_n$ converges to a+b. However, I found it to be difficult to solve $a_n-b_n$ = $a-b$, since the reverse triangle inequality only suggests $\mid a_n-a\...
Serena Chen's user avatar
0 votes
0 answers
68 views

Generalization of triangle inequality in tetrahedron

Let $(a,b)$, $(c,d)$, $(e,f)$ be oppisite edges of the tetrahedron. Our teacher said there's a generalization of triangle inequality: $$ab+cd>ef$$ but I don't know how prove it, and I'm not sure ...
OneLamp's user avatar
  • 395
2 votes
0 answers
89 views

Why is the triangle inequality at the heart of so many proofs in different fields of mathematics?

Question: Why is the triangle inequality at the heart of so many proofs? Discussion In many areas of mathematics, from complex analysis to discrete number theory, and even chaos theory, many theorems ...
Penelope's user avatar
  • 3,211
1 vote
0 answers
75 views

Number of triangles with sides in $\{1,\dots,n\}$

Problem: How many triangles can be formed from a set of $6$ rods of length $1,2,3,4,5,6$ cm taken $3$ at a time? I managed to solve this problem, by applying both the triangle inequalities and testing ...
Cognoscenti's user avatar
0 votes
1 answer
62 views

Can I use triangle inequality to divide infimum of a set?

I am trying to prove $d'(x, y) = \inf\{d(x, y), 1\}$ is a metric on $X$ when $(X, d)$ is a metric space. I am struggling to prove the triangle inequality axiom. I proved it like this: \begin{align*} d'...
Zek's user avatar
  • 309
-5 votes
1 answer
60 views

Maximum length of a side of an acute triangle with a fixed area of 1. [closed]

Is there a maximum length (or at least an upper bound) for a side of an ACUTE triangle with a fixed area? If there is , can it be found?
Simon Ap's user avatar
2 votes
1 answer
100 views

$|x|+|y|\leq 1$ implies $|1-x|+|1-y|\geq 1$

Have a statement that seems true by geometric inspection but I can't seem to prove it using triangle/reverse triangle inequality. $|x|+|y|\leq 1 \implies|1-x|+|1-y|\geq 1$
Magna Wise's user avatar
2 votes
1 answer
88 views

How can I improve my picture proof of Reverse Triangle Inequality?

Diagram beneath reappears on standardized tests IN BLACK AND WHITE with different lengths, letters, and orientation that require students to label in terms of $\vec{b}, \vec{r}$ ( = circle's radius ) ...
user avatar
0 votes
0 answers
171 views

While pictorializing $|x - y| < |x + y|$, how can solely 1 picture simultaneously prove (Reverse) △ Inequalities, $|x-y| ≤ |x|+|y|, |x|-|y| ≤ |x-y|$?

On p. 12, Michael Spivak's Calculus (2008 4 edn) proved $|x + y| ≤ \color{darkgoldenrod}{|x| + |y|}$ (Triangle Inequality). Ibid, exercise 12, p. 16. (iv) ${\color{red}{|x-y|}} ≤ \color{goldenrod}{|x|...
user avatar
1 vote
0 answers
34 views

Show a function is not a norm on $\mathbb{R}^2$ by failing the triangle inequality

I'm trying to show the function $f: \mathbb{R}^2 \to \mathbb{R}$ defined by $f(v) = (\sqrt{|v_1|} + \sqrt{|v_2|}$ is not a norm on $\mathbb{R}^2$ by showing the triangle inequality doesn't hold. That ...
Dylan Hettinger's user avatar
5 votes
1 answer
115 views

Prove that $d(X,Y) = |X\setminus Y| + |Y\setminus X|$ is a distance

I was trying to prove that, for $S$ the power set of $\{1,2,\dots,n\}$, the following function $$d(X,Y) = |X\setminus Y| + |Y\setminus X|$$ is a distance function. I managed to prove positivity and ...
ImHackingXD's user avatar
  • 1,046
0 votes
0 answers
33 views

Average side length of a triangle with perimeter $p$

On the one hand, I think that by symmetry the average side of a triangle with given perimeter $p$ is $\frac{p}{3}$. However (and here I'm probably mistaken), if I look at a side of the triangle, say $...
HappyDay's user avatar
  • 1,037
1 vote
0 answers
67 views

Removing "i" from the triangle inequality?

In my complex analysis class, I have to prove that a limit exists for a function and I think I can use the Triangle Inequality in my proof, but I don't know if it's possible. My question: knowing that ...
ekorel's user avatar
  • 55
0 votes
1 answer
27 views

Given $A \leq X$ and $B \leq Y$, prove $|A - B| + |X - Y| \leq |A - Y| + |X - B|$ [duplicate]

I've been really struggling to prove the following inequality; I'm assuming I'll need to apply the triangle inequality at some point but I can't figure out how to break it down well enough. The ...
Plopleus's user avatar
0 votes
0 answers
52 views

Probability of |BC| < T Given Distances |AB| and |AC|

There are 3 unknown points in Euclidian space namely A,B and C. given distances |BC| and |AC|, what is the probability that |AB| < T. For the 2d space using intersection of the circles and ...
optimum's user avatar
0 votes
2 answers
63 views

Showing that a mapping $d:\{-1,1\}^{\mathbb{N}}\times \{-1,1\}^{\mathbb{N}}\to[0,\infty)$ satisfies the triangle inequality

Define the following mapping $d:\{-1,1\}^{\mathbb{N}}\times \{-1,1\}^{\mathbb{N}}\to[0,\infty)$ by $$d(s^1, s^2)=2^{-\inf\{k\in\mathbb{N}\,: \,s^1_k\neq s^2_k\}}$$ for $s^j=(s^j_1, s^j_2, \ldots), j=1,...
Diffusion's user avatar
  • 5,591
0 votes
1 answer
31 views

Gromov product and distance from a vertex to the opposite line of triangle

I am recently reading definitions of Gromov hyperbolicity. I got stuck on a "trivial" question, that is, given a geodesic triangle $\Delta(x,y,z)$ in any metric space $X$ show that $d(x,[y,z]...
quuuuuin's user avatar
  • 637
0 votes
0 answers
44 views

Do the given lines form a triangle?

Let ABC be an equilateral triangle and let M be a point that does not lie on the circle circumscribed on the triangle. Show that the segments AM, BM and CM can form sides of a triangle. Hint: Check ...
Superunknown's user avatar
  • 2,777
-1 votes
1 answer
75 views

Help me complete the solution to a geometry problem [closed]

This is the solution given in my book but it is incomplete. What will be the next steps to complete the solution . Edit- After simplifying the problem I now have to prove this inequality . ((x+y+z)^...
AB 2008's user avatar
  • 87
2 votes
1 answer
81 views

Strange simple inequality

In an online PDF about continuity the author claim (without proof) that for $h$ small and $f$ continuous we have: $|f(x)-f(x+h)|<\dfrac{|f(x+h)|}{2}$ I try to prove if that statement is true or ...
Vincent ISOZ's user avatar
0 votes
2 answers
65 views

If $a,b,c$ are nonzero natural numbers that verify the relation $a^2+b^2=c^2$, show that $a^n+b^n<c^n$. [duplicate]

PROBLEM If $a,b,c$ are nonzero natural numbers that verify the relation $a^2+b^2=c^2$, show that $a^n+b^n<c^n$, where $n$ is a strictly natural number greater than $2$. WHAT I THOUGHT OF $a^2+b^2=...
IONELA BUCIU's user avatar
  • 1,115
0 votes
0 answers
161 views

Prove that: $|(x+y)(y+z)(z+x)| \le 1.$

Problem complex number inequality: Given $x,y,z$ be complex numbers such that $|x|, |y|, |z| \le 1$ and $|x+y+z| \le 1.$ Prove that $$|(x+y)(y+z)(z+x)| \le 1.$$ I see it on AOPS here. Here is my ...
Nguyễn Thái An's user avatar
3 votes
0 answers
74 views

Confirming an answer using the Triangle Inequality for complex numbers

I was wondering if someone can point me in the right direction in solving the following complex number problem. The problem is listed below. The variable complex number z satisfies |z − 2 − i| = 1. ...
Stephan's user avatar
  • 469
6 votes
1 answer
104 views

$\ \forall x_1,x_2,...,x_n \in \mathbb{R} (x_i\not=x_j)$ in the range of $[-1,1]$ prove:$\sum_{i=1}^{n}\frac{1}{\Pi_{k\not=i}|x_k-x_i|}\ge2^{n-2}$

$\ \forall$ $x_1,x_2,...,x_n$ $\in \mathbb{R}$ $(x_i\not=x_j)$ in the range of $[-1,1]$ prove : $$\sum_{i=1}^{n}\frac{1}{\Pi_{k\not=i}|x_k-x_i|}\ge2^{n-2}$$ my attempt : $$p(x) = \sum_{i=1}^{n}\left(...
farnood gholampoor's user avatar
0 votes
1 answer
62 views

$f(x)=x^n+a_{n-2}x^{n-2}+...+a_1x+a_0\in \mathbb{R}[X]$. prove$\ \exists$ $i\in[1,...,n]$ so that : $|f(i)| \ge \frac{n!}{\binom n i}$

$f(x)=x^n+a_{n-2}x^{n-2}+...+a_1x+a_0\in \mathbb{R}[X]$. prove$\ \exists$ $i\in[1,...,n]$ so that : $$|f(x)| \ge \frac{n!}{\binom n i}$$ my attempt : i used lagrange interpolation and compared $x^n$ ...
farnood gholampoor's user avatar
0 votes
0 answers
18 views

Bound from below with Triangle inequality

This might be a fairly easy question but I am a bit stuck. I am trying to bound from below the following quantity \begin{equation} |A x \cdot x|^{\frac{1}{2}}, \end{equation} where $A$ is a complex-...
Jason Curran's user avatar
0 votes
1 answer
40 views

Complex integration - showing that the arc integral vanishes using the estimation lemma

I came across the following integral: $$\int_{- \infty}^{\infty} \frac{x^2}{((x-t)^2 + \delta^2 )^2((x + t)^2 + \delta^2)^2} \textrm{d}x.$$ I understand that if we turn this into a complex integral ...
Jessica Barr's user avatar
4 votes
4 answers
152 views

Prove: $\sum\limits_{cyc} \sqrt{9a^2+(a+b+c)^2} \ge \sqrt{12(a^2+b^2+c^2)+14(a+b+c)^2}$ with $a,b,c>0.$

Let $a,b,c>0.$ Prove that $$ \sqrt{9a^2+(a+b+c)^2}+\sqrt{9b^2+(a+b+c)^2}+\sqrt{9c^2+(a+b+c)^2} \ge \sqrt{12(a^2+b^2+c^2)+14(a+b+c)^2}$$ I see it on Facebook here. I tried Mincopxki, but it doesn't ...
Nguyễn Thái An's user avatar
-1 votes
1 answer
49 views

Inequality in a random triangle [closed]

I'm studying the gravitationnal field produced by the Sun at a point at the surface of the Earth and at the center of the planet. I've reduced the situation to a geometry problem but I'm stucked with ...
Ben's user avatar
  • 3
0 votes
0 answers
41 views

Proof of triangle inequality (where d is a metric described in the question)

Let $x, y, z$ be some three (real-valued) time series. Prove that $$\sqrt{1/2-\rho(x,z)/2} \leq \sqrt{1/2-\rho(x,y)/2} + \sqrt{1/2-\rho(y,z)/2}$$ After this assuming that two time series cannot be ...
nura khazhimurat's user avatar
0 votes
0 answers
295 views

Why is the triangle inequality in a weighted graph always satisfied? Why is this not a counter-example?

I am wanting to know why the triangle inequality is always satisfied in a weighted graph. In particular, I want to know why the attached graph (please click the link weighted K_5 below) is not a ...
Emily's user avatar
  • 1

1
2 3 4 5
7