I'm stuck in a property when I was reading proof of a theorem about polyhedral theory:
Every full dimensional, bounded and non-empty polyhedron in $\mathbb R^n$ has at least $n+1$ vertices.
A vertex is a point of a polyhedron such that we cannot write it as $\lambda x+(1-\lambda)y$ such that $\lambda \in (0,1)$.
Can anyone give me a hint to prove this property? I think I should consider the fact that this polyhedron is convex hull its vertices and with induction I should prove that convex hull of every $k$ elements is at most a $k-1$ dimensional space. But it doesn't make sense to me also these properties. Could you please help me to understand these fact truly?
Since you already know that a polyhedron is the convex hull of its vertices, the proof can proceed like this. Suppose the vertex set is $v_1, \dots, v_m$ with $m\le n$. The vectors $v_1-v_m,\dots, v_{m-1}-v_m$ span a linear subspace $W\subset \mathbb{R}^n$ of dimension at most $m-1$. Therefore, the points $v_1, \dots, v_m$ lie in the affine subspace $W+v_m$ of dimension at most $m-1 < n$. Since $W+v_m$ is a convex set, the convex hull of $v_1, \dots, v_m$ is contained in $W+v_m$ and therefore has empty interior.