Prove that every convex region is simply connected
Could anyone help?
Prove that every convex region is simply connected
Could anyone help?
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.