# Tagged Questions

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### Proof of triangle inequality for $d(x,y)=\sqrt{\lvert x-y\rvert}$

There is this problem that says: show that $d(x,y)=\sqrt{\lvert x-y\rvert}$ is a distance function on $\mathbb{R}$, and I am unable to proof the triangle inequality for this? any suggestion I look ...
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### Show that $d^\ast$ is a metric

For $x$ and $y$ in $R$, let $d(x,y)$ be a metric. Show that $$d^\ast(x,y)=\frac{d(x,y)}{1+d(x,y)}$$ is also a metric. It is fairly straightforward to show that $d^\ast(x,y)=0$ if $x=y$ ...
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### Find Weight for minimum Manhattan Distance

Let's say, I have three points $(1, 4)$, $(4, 3)$ and $(5, 2)$. I need to find weight $w_1$ and $w_2$ so that the point $(1, 4)$ be the centroid of the points in ...
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### A Particular Metric: $(\mathbb{R}^2,d_2)$

I'm trying to show that the metric $(\mathbb{R}^2,d_2)$ is indeed a metric. Here $$d_2((x_1,y_1),(x_2,y_2))=\max\{\lvert x_1-x_2\rvert, \lvert y_1-y_2\rvert\}.$$ I've got everything up to the ...
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### Prove triangle inequality

I want to prove that $d(x,y) = 1- \sum_i {\min(x_i, y_i)}$ where $\sum_i {x_i} = \sum_i {y_i} =1$ and $\forall i: x_i, y_i \geq 0$ satisfies the triangle inequality. The domain of $d$ therefore is ...
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### Metric on the set of CDFs with finite p-th moment

Let $\mathcal{F}_p$, $p \ge 1$, be the set of all cumulative distribution functions of real valued random variables whose $p$-th moment is finite. I'm looking for a metric on $\mathcal{F}_p$ and ...
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### Metric on a set

Can someone provide a hint for solving the following. Show that $d:(R^{\infty})^2\to R_+$ is a metric. $$d(x, y)=\sqrt{\sum_{i=0}^{\infty}{(x_i-y_i)^2}}$$ I need a hint for showing that $d$ ...
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### Euclidean Metric satisfying the Triangle Inequality - Is there missing details in the proof given here?

The image below comes from the book Geometry and Topology by Miles Reid and Balazs Szendroi. They prove the Triangle Inequality, which is stated below $(2)$. I am happy with the proof of the ...
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### $|f(x)-f(y)|\le(x-y)^2$ without gaplessness

If $|f(x)-f(y)|\le(x-y)^2$ for all $x,y\in\mathbb R$, then it's easy to show that $f'=0$ everywhere, and the mean value theorem implies that that means $f$ is constant. If there were a gap in the ...
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### How to prove the strong triangle inequality?

I'm trying to prove that p-adic space is a metric space(also a ultrametric space), but I find it difficult to prove the triangle inequality. So if one can prove the strong triangle inequality, then ...
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### How to prove this is a metric?

I have 2 distance functions $d(x,y)=|x^2-y^2|$ and $d(x,y)=|x^3-y^3|$ and I am trying to prove that they are metrics on $\mathbb R$, or give a counterexample that they are not metrics on $\mathbb R$. ...
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### General Triangle Inequality, distance from a point to a set

I am trying with no luck to prove: Let (X,d) be a metric space and A a non-empty subset of X. For x,y in X, prove that d(x,A) < d(x,y) + d(y,A)
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### unit ball question

If $1\leq p<q$, show that the unit ball $l_{n}^p(\mathbb{R})$ is contained in the unit ball $l_{n}^q(\mathbb{R})$. Well the definition of $l_{n}^p(\mathbb{R})$ is that for ...
This is probably very simple, but for some reason I don't seem to see why if $\forall x \in \mathbb R^n, \|x-\phi(x)\|>c$ for some $c>0$ then $\|x-y\|>\|\phi(x)-\phi(y)\|$, where $\phi$ is ...