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Let $G$ be the annulus determined by the inequalities $2 < |z| < 3$. This is a connected open set.

Find the maximum number of horizontal and vertical segments in $G$ needed to connect two points of $G$.

I am thinking the maximum number should occur at the endpoints, but since we have an open annulus, the endpoints can not be obtained...Any help?

Thanks~

Matata
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  • So draw some pictures. What is your educated guess? – Ted Shifrin Dec 18 '17 at 03:39
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    @TedShifrin Does that occur when we wan to connect some point near $(0,3)$ to $(0,-3)$? – Matata Dec 18 '17 at 03:50
  • As Ted Shifrin suggested, draw some pictures. Could you, without loss of generality, take one of the points to be in the upper right quarter of the annulus, i.e., in ${ z \in G \mid \operatorname{Re} z \geq 0 \text{, } \operatorname{Im} z \geq 0}$? I believe so. I think you'll see that your pictures need a certain geometric figure included to help you get all the different cases. – Mark Twain Dec 18 '17 at 03:54
  • A First Course in Complex Analysis by Matthias Beck, Gerald Marchesi, Dennis Pixton, and Lucas Sabalka? – BCLC Jul 31 '18 at 02:17

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