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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. – 2017-02-21
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    but how big is it and where do i place it on a graph?? @Alephnull – 2017-02-21
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    hint: 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. – 2017-02-21
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    Draw the boundary of the closed ball, that is $|x|+|y|=1$. What is inside is the open ball. – 2017-02-21
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    @OPFragster Sorry, I'm not that kind of girl. – 2017-02-21

1 Answers 1

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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: 1

Using Wolfram alpha: https://www.wolframalpha.com/input/?i=plot+%7Cx%7C+%2B%7Cy%7C+%3D1