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If I am provided with function such as this $$f(x) =|x_1|+|x_2|,$$ how do I know what kind of level set it generates? Please, generalize the result. Similarly, for $$f(x) =−(x_1−2)^2+|x_2|+ 1.$$

I don't know how to determine which function generates superlevel, sublevel, or level set.

reyna
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Jimjamlorde
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  • So, I know that |x1|+|x2| = C and similarly −(x1−2)2+|x2|+1 = C and we use different values of C to generate the level set. But I don't know how exactly is the level set produced. and what kind of level set is what is produced. Please guide @viktor – Jimjamlorde Oct 05 '20 at 07:00
  • I recommend making case distinctions ($x_1,$ positive, negative and so one) and then rearranging to solve to the values the other variable can attain. – ViktorStein Oct 05 '20 at 07:23
  • Can you explain what it means to "generate superlevel sets" (or sublevel or level sets, for that matter)? – ViktorStein Oct 06 '20 at 14:33

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Hints: Let's look at the level set for 1 for the first function. If $x_1$ and $x_2$ are positive, we can write $x_1+x_2 = 1$. Recognise this as the equation of a straight line and draw this in the positive quadrant of the coordinate system. Now continue for $x_1 < 0, x_2 > 0$ and the other two cases.

The definition of a level set I know is $$ N(f, a) := \{ x \in \mathbb R^2: f(x) \le a\}. $$ Thus the level sets of the first functions are those tilted squares, not just their boundary.


With a similar case distinction to the first function, can you show that for $a \ge 1$, the level set $N(g,a)$ for the second function $g$ consists of two parabolas?
ViktorStein
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