I have the following exercise in a course of convex geometry by Hug and Weil :
Let $P$ be a simplex with $n+1$ vertices $x_0,\dots, x_n$ in $\mathbb{R}^n$, that is, $P$ is the convex hull of $n+1$ affine independent points $x_0,\dots, x_n$. Let $E_i$ be the affine hull of $\{x_0\dots,x_n\}\setminus\{x_i\}$ and $H_i$ the closed half space bounded by $E_i$ containing $x_i$. Show that $$P=\bigcap_{i=0}^n H_i.$$
For the first part it is easy to see that $P$ is included in each $H_i$ and hence is included in their intersection. For the other inclusion, I would take $x\in \bigcap_{i=0}^n H_i$ and write $x=\sum_{i=0}^n \alpha_ix_i$ with $\sum_{i=0}^n \alpha_i=1$. Since $x\in H_i$, I am sure that $\alpha_i$ must be nonnegative, however I do not know how to prove the latter.
Can someone help me? Thanks a lot!
Suppose $ x\not \in P$. Write $x= \sum_k \lambda_k x_k$ where $\lambda_k$ are the barycentric coordinates of $x$ with respect to $x_0,...,x_n$. Note that since the $x_k$ are affinely independent, the coordinates are unique.
Since $ x \not \in P$, we have $\lambda_{k'} < 0$ for at least one $k'$. Then $x \not \in H_{k'}$.
Hence $P^c \subset (\cap_k H_k)^c$.
Addendum:
You can define $H_i$ as either $\{ (1-\lambda)\sum_{k \neq i} \lambda_k x_k + \lambda x_i| \sum_k \lambda_{k \neq i} =1, \lambda \ge 0 \}$ or equivalently $\{ \lambda x_i + (1-\lambda) e | \lambda \ge 0, e \in E_i \}$.