For points $p = (p_1, p_2)$ and $q = (q_1, q_2)$ in $\mathbb{R}^2$ define:

$d_V(p,q) = \begin{cases}1 & p_1\neq q_1 \ or\ |p_2 - q_2|\geq 1 \\ |p_2 - q_2| & p_1= q_1 \ and\ |p_2 - q_2|< 1 \end{cases}$

Show that $d_V$ is a metric.

My Unfinished Solution:

So there are three states:

$1- \ p_1= q_1 \ and\ |p_2 - q_2|\geq 1 \\ 2- \ p_1= q_1 \ and\ |p_2 - q_2|< 1 \\ 3- \ p_1\neq q_1$

The first and second criterion of a metric space are easy to prove. But I can't gather all the three above-mentioned states for $(p,q,r)$ to prove metricness of $d_V$ for the third criterion, i.e., $d(p, q) + d(q, r) \geq d(p, r)$ for all $p,q,r \in \mathbb{R}^2$.

Thank you.


$\forall p,q : \ d_V(p,q) = |p_2 - q_2| \mathrm \ or \ 1 \ $ and $\ d_V(p,q) \le 1$

Specially,if $ d_V(p,q) =1$ then $d_V(p,q)\le |p_2 - q_2|$

CASE I: $d_V(p,r)=|p_2 - r_2|\ \le |p_2 - q_2|+|q_2 - r_2|$

(Remark: $|p_2 - r_2|\ \le |p_2 - q_2|+|q_2 - r_2|$ is from basic inequation in R: $|a+b|\le |a|+|b|$)

$ \ $

If $|p_2 - q_2|<1$ , $|q_2 - r_2|<1,p_1=q_1,q_1 = r_1$

Then $d_V(p,q)+d_V(q,r)= |p_2 - q_2|+|q_2 - r_2|\ge d_V(p,r)$

$ \ $

Other cases, at least one of equations :$\ d_V(p,q) =1 $ , $d_V(q,r)=1$ holds.

Hence, $d_V(p,q)+d_V(q,r)\ge 1\ge d_V(p,r)$

$ \ $

(Remark: Follow the above argurment, get that $d_V(p,q)+d_V(q,r) = |p_2 - q_2|+|q_2 - r_2| $ or $ \ge 1$)

CASE II: $d_V(p,r)=1 \le |p_2 - r_2|\le |p_2 - q_2|+|q_2 - r_2|$

By the similiar arguement did in CASE I,find $d_V(p,q)+d_V(q,r)\ge d_V(p,r)$

  • $\begingroup$ Thank you very much for your answer. But still it seems to have problems: Supposing CASE I holds for $(p,r)$, it doesn't mean that each of $(p,q)$ and $(q,r)$ satisfies CASE I. In 4th line you supposed that all $p,q,r$ satisfy CASE II, and in 6th line you supposed $(p,r)$ satisfies CASE I but $(p,q)$ and $(q,r)$ satisfy CASE II. Am I right? $\endgroup$
    – L.G.
    Jan 24 '15 at 8:11
  • $\begingroup$ @Ali.E. I put much more details. $\endgroup$
    – Brian
    Jan 24 '15 at 8:45
  • $\begingroup$ @Syuizen: Would you please guide me how does a open 'ball' look like in this metric? I mean, e.g., for an open 'ball' for taxicab metrican is an open diamond, centered at $p$, with distance $\epsilon $ from $p$ to the corners. Thank you. $\endgroup$
    – user210902
    Feb 1 '15 at 1:01

Suppose we have three distinct points $x_1 = (p_1,q_1), x_2 = (p_2, q_2), x_3 = (p_3, q_3)$ in the plane (in proving the triangle inequality we can always assume that the three points are distinct; when two or more are equal, it already follows from the other axioms)

We want to show $d_V(x_1, x_3) \le d_V(x_1, x_2) + d_V(x_2, x_3)$.

It's clear from the definition that $d_V(x_1, x_3) \le 1$ always (as is any value of $d_V$). So we can assume (or we are done) that none of the two distances on the right hand side is $1$ and their sum is $<1$ as well. This can only happen if $p_1 = p_2$ and $d_V(x_1,x_2) = |q_1 - q_2| < 1$ and $p_2 = p_3$ and $d_V(x_2, x_3) = |q_2 - q_3| < 1$.

So in particular, $p_1 = p_3$. We know that $|q_1 - q_3| \le |q_1 - q_2| + |q_2 - q_3| = d_V(x_1, x_2) + d_V(x_2, x_3) < 1$ (by assumption), using in the first step the usual inequality for $|\cdot|$, and so $d_V(x_1, x_3) = |q_1 - q_3| \le d_V(x_1, x_2) + d_V(x_2, x_3)$ and we are done.


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