My Question is:
Sketch the unit ball $B(0, 1)$ in $\mathbb{R}^2$ equipped with the following norm: $\|(x, y)\| =|x|+|y|$
I'm semi confident in this topic but cant seem to find the right graph to sketch so any help will be appreciated.
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It is a diamond. – Jacob Wakem Feb 21 '17 at 01:35
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but how big is it and where do i place it on a graph?? @Alephnull – Feb 21 '17 at 01:40
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4hint: the result is ${(x,y)||x|+|y|\leq1}$. How to draw this? Assume $x,y>0$, draw it, and by symmetry, you know what the whole picture is. – Eric Feb 21 '17 at 01:42
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1Draw the boundary of the closed ball, that is $|x|+|y|=1$. What is inside is the open ball. – Masacroso Feb 21 '17 at 01:45
1 Answers
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Using the metric, you can sketch the outline simply by solving for all possible equations which have $d(x,y) =1. $
As such you get four equations:
$|y| +|y| = 1$ yields:
$$|y| = |x| -1$$
By symmetry, these equations only have to be solved for $y \leq 0$. Which gives:
$$ y= x -1 \text{ if } 0 \leq x \leq 1\\
y= -x -1 \text{ if } -1 \leq x<0$$
Which in turn gives:
Using Wolfram alpha: https://www.wolframalpha.com/input/?i=plot+%7Cx%7C+%2B%7Cy%7C+%3D1
