Is the following set $A=\{x\in\mathbb R^{n}:||x-z||\le||x-y||,\forall y\in K, K\subset\mathbb R^{n} \}$ convex? I need to prove it according to definition.
We have $x_{1},x_{2}\in A, \lambda\in (0,1)$ and we need to show that $\lambda x_{1}+(1-\lambda) x_{2}\in A$
Since $x_{1},x_{2}\in A$, that is, $||x_{1}-z||\le||x_{1}-y||, ||x_{2}-z||\le||x_{2}-y||$ and we need to prove $||\lambda x_{1}+(1-\lambda) x_{2}-z||\le||\lambda x_{1}+(1-\lambda) x_{2}-y||$
$||\lambda x_{1}+(1-\lambda) x_{2}-z||=||\lambda (x_{1}-z)+(1-\lambda) (x_{2}-z)||\le\lambda||x_{1}-z||+(1-\lambda) ||x_{2}-z||\le \lambda||x_{1}-y||+(1-\lambda)||x_{2}-y||$ Now, I don't know the next steps.