# straightforward way to determine if this set is convex?

straightforward way to determine if this set is convex? $Z=\left\{x\in\mathbb{R}^2:3x_1^4-x_1x_2+x_2^4\le x_2,x_1>2,x_2>2\right\}$

I know I can try by manipulation of linear combination of two points of this set Is there any simpler way than using the definition $$f(x_1, y_1)<0 \wedge f(x_2, y_2)<0 \Rightarrow f(t x_1+(1-t)x_2, t y_1+(1-t)y_2)<0\;$$ $$\text{ for }\; 0 < t < 1?$$

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maybe you can use something like "the intersection of convex sets is again convex" –  Quickbeam2k1 Mar 5 '13 at 20:00
I think this is not convex –  lizusek Mar 5 '13 at 20:01
did you try to differentiate twice the function that determines your first relation? –  Quickbeam2k1 Mar 5 '13 at 20:05
Oh sets? That must be [set-theory]! –  Asaf Karagila Mar 5 '13 at 20:06
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## 2 Answers

The set is empty, because $$(x_1+1)x_2\le 2x_1x_2\le x_1^2+x_2^2<3x_1^4+x_2^4$$ in the range $x_1,x_2>2$. The empty set is usually considered convex.

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yes, I have seen that this is empty, but shoudn't we take the sum of these two sets? I think this meant to be sum –  lizusek Mar 5 '13 at 20:15
@cf16 No. The notation $\{x: P,Q,R\}$ always means the set of points $x$ that satisfy the conditions $P$ and $Q$ and $R$. If this notation is misused here to mean or, then the set is still not convex: it contains $(0,10)$ and $(10,0)$, but not $(5,5)$. –  user53153 Mar 5 '13 at 20:19
yes, I have also another example. I think I will show both explanations, in any case –  lizusek Mar 5 '13 at 20:34
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yes, so if this is empty, this is convex by definition. if this was meant to be sum of two sets then my solution is take one point that belongs to set of function solutions and one from other set, we see $f(X1)<0$ so it is solution of $f$, and $X2=(2.2,2.2)$ so it belongs to the second set, now $f(\alpha X_1+(1-\alpha)X_2)>0$ so it doesn't belong to f solutions and also not to the second set as $X=(1.105,1.35)$

    x        y       f
X1  0,01    0,5 -0,44249997
X2  2,2     2,2   86,6624

alfa   0,5  0,5

X   1,105 1,35   4,952462402

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