I've just come across an exercise in complex analysis that: Let $U\subset \mathbb C^n$, $n>1$ be a domain and $K\subset\subset U$ be a compact geometrically convex subset. If $f\in \mathcal O(U\setminus K)$ then prove that $f$ extends to be holomorphic in $U$.
Hint: Find a nice point on $\partial K$ and try extending a little bit. Then make sure your extension is single-valued.
I would like to see a proof of this using this hint.. Thanks