discuss convexity of the following set ?
$$M= \{(x,y)∈\Bbb R^2 : x^2+y^2≥a^2 ,x^2+y^2≤b^2 ,x>0,y>0\} $$
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1And what did you do, exactly? – mookid Oct 30 '14 at 22:13
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What exactly is your question? – daOnlyBG Oct 30 '14 at 22:14
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2Have you tried to plot the region? – mfl Oct 30 '14 at 22:14
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i think this set is not convex but how can i prove that ? but i found a theory that the sum of two convex set is convex too – Belal Elprmawy Oct 30 '14 at 22:17
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mfl's suggestion to plot the region is a very good idea. – Jay Oct 30 '14 at 22:24
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thanks for all your efforts – Belal Elprmawy Oct 30 '14 at 22:40
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For this to be a convex set, you need to be able to draw a line $L$ between any two points $(x_1, y_1), (x_2,y_2) \in M$ such that every point $(x,y) \in L$ is also in $M$.
In this case, it's obvious that the set $M$ can only be convex if $a = 0$ and $b>0$ (Naturally, $b>a$).
If $a>0$, there will always exist a small $\epsilon > 0$ such that a line $L$ from $(a,\epsilon), (\epsilon, a)$ will always pass through your "donut hole".
Convex sets are important in discussing the uniqueness of solutions in optimization problems --which is what I'm assuming you're studying given the tags you've put on this question. Consider reading the wiki on this topic, the images on the right sidebar illustrate my point: http://en.wikipedia.org/wiki/Convex_set
Edit: Typo(s)
