1
vote
1answer
17 views

Show that $d(u,v)=\exp(-\max\{j\ge 0, u_k=v_k \space\mbox{for}\space 0\le k\le j\})$ is a distance over $E=\Bbb{R}^\Bbb{N}$.

Let $E=\Bbb{R}^\Bbb{N}$, $u=(u_k)_{k\in\Bbb{N}}$ and $v=(v_k)_{k\in\Bbb{N}}$. Define $$ d(u,v) = \left\{ \begin{array}{ll} \exp(-V(u,v)) & \mbox{if}\quad u\ne v \\ 0 ...
2
votes
1answer
19 views

Is $d(i,j) = 1-\textrm{corr}(i,j)$ a metric?

I need to make sure that this function is a metric: $d(i,j) = 1-\textrm{corr}(i,j)$ where $\textrm{corr}(x,y)$ is the Pearson correlation coefficient which ranges from $[-1,1]$. With this scaling I ...
1
vote
1answer
30 views

Proving that $d(x,y) =\max_i{\lvert x_i - y_i \rvert }$ is a distance in $\mathbb{R}^2$

I was asked to prove that $d(x,y) =\max_i{\lvert x_i - y_i \rvert }$ is a distance function in $\mathbb{R}^2$. I've got myself stuck with proving the triangle inequality. Can someone give me an hint ...
0
votes
0answers
27 views

Proof metric space with distance function

Thats the first time i have to do such an proof but don't know how, never seen or done this before. Especially (iii). Let $X$ be the Set of all complex sequences. $$ d((a_n),(b_n)) := ...
1
vote
3answers
66 views

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 ...
2
votes
2answers
101 views

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 $x=y$ ...
1
vote
2answers
110 views

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 ...
1
vote
1answer
65 views

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 ...
3
votes
1answer
115 views

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 ...
2
votes
0answers
73 views

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 ...
0
votes
2answers
69 views

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 $d$ ...
2
votes
1answer
130 views

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 ...
10
votes
2answers
142 views

$|f(x)-f(y)|\le(x-y)^2$ 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 ...
0
votes
1answer
672 views

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 ...
2
votes
0answers
96 views

Inner product and inequalities

Suppose $p:[0,1]\to \mathbb C$ is a curve where $p(t)=u(t)+iv(t)$ and $u,v$ are smooth functions of $t$. Why then is $$\left(\int_0^1 \langle \dot{p},\dot{p}\rangle^{1\over 2} dt\right)^2\le \int_0^1 ...
1
vote
3answers
154 views

Proving this function is a metric

This is a space $S$ that consists of the set of all sequences of real numbers and $x=(x_1,x_2,x_3,...), y=(y_1,y_2,y_3,...)$ etc. and the metric $d$ is defined as $$d(x,y)=\sum_{i=1}^\infty ...
2
votes
2answers
2k views

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 $\mathbb R$. ...
1
vote
1answer
1k views

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)
1
vote
1answer
192 views

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 ...
1
vote
1answer
84 views

Simple inequality question

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 ...
0
votes
2answers
133 views

Why do these inequalities in metric spaces hold?

The other day I stumbled across some inequalities regarding properties of metric spaces. I'm curious to see a proof of why it holds. Suppose $(X,\rho)$ is any metric space. For a given $\epsilon\gt ...
3
votes
2answers
2k views

Proof of Triangle Inequality on $(\mathbb{R}^n, d_p)$

I have to prove the triangle inequality $(|x_1 - z_1|^p + |x_2 - z_2|^p)^{1/p} \leq (|x_1 - y_1|^p + |x_2 - y_2|^p)^{1/p} + (|y_1 - z_1|^p + |y_2 - z_2|^p)^{1/p}$ for $p \geq 1$ on $\mathbb{R}^2$. ...