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What do the unit balls of this metric look like for radii 1/2, 1 and 3/2 please ?

Radar screen metric is $d(x,y):= \min(1, \|x - y\|_2)$, where the subscript $2$ after the last $\|$ indicates it's over $\mathbb{R}\times\mathbb{R}$.

Thanks

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HINT:

  • The metric $d$ is identical to the usual metric when $\|x-y\|\le 1$; what does that tell you about the $d$-balls of radii $\frac12$ and $1$?

  • Now let $x\in\Bbb R^2$ be arbitrary. The open $d$-ball of radius $\frac32$ centred at $x$ contains those points $y\in\Bbb R^2$ such that $d(x,y)<\frac32$. Can you find any point $y\in\Bbb R^2$ such that $d(x,y)$ is not less than $\frac32$?

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  • $\begingroup$ So the balls of radii 1/2 and 1 are concentric balls at x whereas the ball of radius 3/2 (ie 1.5 ) is off the screen as its out of range ? ...ie its bigger than min (1, ||x-y||) ? $\endgroup$
    – Des
    May 15, 2013 at 5:16
  • $\begingroup$ @Des: Since $\min\{1,\|x-y\|\}$ is not a single number, but rather (for fixed $x$) a function of $y$, it doesn’t make sense to say that some ball is bigger than that. Look at the definition of $B\left(x,\frac32\right)$: it’s $\left\{y\in\Bbb R^2:d(x,y)<\frac32\right\}$. What subset of $\Bbb R^2$ is that? $\endgroup$ May 15, 2013 at 5:19
  • $\begingroup$ Sorry .....you have lost me. The question said "draw the unit balls of radii 1/2, 1 , 3/2 "....which in itself seems contradictory cos if its a unit ball how can it have a radius other than 1 ? $\endgroup$
    – Des
    May 15, 2013 at 7:01
  • $\begingroup$ @Des: The question is internally inconsistent. The unit ball must indeed have radius one, so it’s impossible to draw the unit ball of radius $\frac12$ or $\frac32$. Some part of the question must therefore be in error. It’s most likely that the word unit is the mistake, and that you’re intended to draw the balls of radii $\frac12,1$, and $\frac32$. $\endgroup$ May 15, 2013 at 15:09
  • $\begingroup$ You are correct ....i checked with the author and it is a mistake ....should just say draw the balls of radii 1/2,1 and 1.5 ....that being the case they would just be concentric circles of radii 1/2, 1 and 1.5 in R 2 ....right ? $\endgroup$
    – Des
    May 18, 2013 at 4:43

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