The convex hull of a certain set contains the origin in its interior. For  $c_1,\dots,c_n\in \mathbb{R}^l$ fixed, I am trying to show that
If $0\in int \{\sum_{i=1}^n\alpha_ic_i:\alpha_i\ge 0,\ \sum_{i=1}^n \alpha_i=1\}$, then  $0\in int \{\sum_{i=1}^n\alpha_iv_{i}:\alpha_i\ge 0,\ \sum_{i=1}^n \alpha_i=1\}$ for all $v_{1},\dots,v_{n}\in \mathbb{R}^l$ with $ \sum_{i=1}^n||c_i-v_{i}||<\varepsilon$,  if $\varepsilon$ is small enough.
Would appreciate some help 
 A: Let $C_0 = \{c_1, \dots, c_n\}$, and let $C$ be the convex hull of $C_0$ in $\mathbf{R}^l$. Since $0 \in \operatorname{int} C$, there is some simplex $D$ with vertices $d_0,\dots, d_l$ all in $C$ such that $0$ is in the interior of $D$. (For example, $D$ is a tetrahedron when $l = 3$.)
The point $0$ can be written uniquely as a convex combination $\sum \beta_j d_j$ with $\beta_j > 0$.
Now write each $d_j$ as a convex combination $\sum_i \alpha_{j,i} c_i$, and let $w_j = \sum_i \alpha_{j,i} v_i$. As $v_i \to c_i$, we have $w_j \to d_j$. For $v_i$ close enough to $c_i$, the $w_j$'s will be affinely independent since the $d_j$'s are (i.e., $\det(w_1-w_0,\dots,w_l-w_0)\ne 0$).
Now write $0$ (uniquely) as an affine combination $\sum \gamma_j w_j$. Since $w_j \to d_j$ and the coefficients $\gamma_j$ are continuous functions of the $w_j$'s, we have $\gamma_j \to \beta_j >0$. Once all the $\gamma_j$'s are positive, $0$ will be in the interior of the simplex generated by the $w_j$'s.
A: Let I be a closed and bounded interval, and f (x): I-> I is a continuous function. Show that there exists x element of I such that f (x) = x proves that x is not necessarily unique
