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Prove that every convex region is simply connected

Could anyone help?

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    You have to prove two things: that it's path connected, and that every continuous image of $S^1$ can be shrunk to a point (or some equivalent statement). Can you at least do the first?2012-05-01
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    Is it true that a convex region is star-like? If this is true, it could help.2012-05-01
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    In, fact, this is true, once every point can be joined by a segment to another one. So, choose one and use this point to construct a homotopy. I guess this could wortk.2012-05-01

1 Answers 1

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Choose a point $z_0\in R$ (which denotes the region). Choosing any other point $z\in R$, the segment joining these two is contained in $R$, by convexity. Then, you are able to construct a linear contraction of $R$ to this $z_0$:

$$C:[0,1]\times R\rightarrow R$$ $$(t,z)\mapsto z_0+t(z-z_0)$$

Therefore, any closed path $\gamma:\mathbb{S}^1\rightarrow R$ can be shrunken to $z_0$ by composing the path with the contraction.