Consider linear subset $H$ of a normed space $X$ such that there exists a $c > 0$ and $x_0 \in X$ with
$ \Vert x−x_0\Vert \ge c$ for all $x \in H$,
every $x \in X$ is represented in the form $x = tx_0 + y$ with $t \in \mathbb{R}$ and $y \in H$.
Now I am supposed to show that this set is convex. I tried some attempts, but it didn't work out. I know that I need to verify for $\lambda \in (0,1)$ and $x,y \in H$ that $\lambda x + (1-\lambda)y \in H$.