I have convex the set $C:=C(x_1,\dots,x_n) \in \mathbb{R}^n$ of convex combinations of $x_i$'s. I know that there exists an $x_i$ such that $\Vert x_i \Vert > 0$ and $ 0 \notin \mathring{C}(x_1,\dots,x_n) = \{\sum_{i=1}^{n}\lambda_i x_i : \lambda_i \in (0,1) \text{ and sum up to 1}\}$. I also know that \begin{align*} \exists y \in C \>\forall c \in \mathring{C} : \> \langle y , c \rangle > 0. \end{align*} Now I am supposed to show that $\langle y , x_i \rangle > 0$ holds.
Can somebody give me a hint for this task?
The result seems to be wrong.
Take the plane, i.e. $\mathbb R^2$ with $x_1 = (1,0)$ and $x_2=(0,1)$. You have $\Vert x_1 \Vert = 1 \neq 0$. Now $y = x_2$ belongs to $C(x_1,x_2)$ and $\langle y, c \rangle >0$ for all $c \in \mathring{C}(x_1,x_2)$. Also $0 \notin \mathring{C}(x_1,x_2)$.
However $\langle y , x_1 \rangle = \langle x_2, x_1 \rangle = 0$.