# 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 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 ...
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1 vote
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### 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 ...
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### 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 ...
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### 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 ...
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1 vote
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### 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 ...
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### 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 ...
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1 vote
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### 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 ...
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### 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 $E$, $d(A, B) \leq d(A, C) + d(C, B)$?
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### 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....