I have to prove or disprove that $$ \left\{x \in \Bbb R^n : \sum_{i=1}^n x_i^2 = 1\right\}\text{ is convex.} $$ I already know that it is not convex, so a disprove is the right way.
I know that I gotta use the following for a prove: $$ \text{A set }\Omega\subset\Bbb R^n\text{ is convex if }\alpha x+(1−\alpha)y\in\Omega,\forall x,y\in\Omega\text{ and }\forall\alpha\in[0,1]. $$ For a disprove I could take two points and show that the above condition is not met. But what confuses me is that the statement also depends on the value of $n$. If I choose e.g. the two points $\sqrt{\frac{7}{10}}$ and $\sqrt{\frac{3}{10}}$, how do I could use them to disprove it?