Consider $n+1$ points, $\{v_{0},v_{1},\ldots, v_{n}\}$ in $\mathbb{R}^{m}$ such that, $m\geq n$. These $n+1$ points are geometrically independent. This means that if for some reals $\{\lambda_{0},\lambda_{1},\ldots,\lambda_{n}\}$
$$\sum_{i=0}^{n}\lambda_{i}v_{i}=0 \qquad \sum_{i=0}^{n}\lambda_{i} = 0$$
implies $\lambda_{i} = 0 \quad \forall \quad i$.
Consider an $n$-dimensional plane in $\mathbb{R}^{m}$ passing through all the points $\{v_{0},v_{1},\ldots, v_{n}\}$. This consist of all the points of the form $$ \sum_{i=0}^{n} \lambda_{i}v_{i} \quad s.t. \quad \sum_{i=0}^{n}\lambda_{i} = 1$$
It's easy to see that we can associate to every point $v$, of the plane, a collection of $n+1$ reals $\{\lambda_{0},\lambda_{1},\ldots,\lambda_{n}\}$, such that $v = \sum_{i=0}^{n}\lambda_{i}v_{i}$ and $\sum_{i=0}^{n}\lambda_{i}=1$ and that this association is unique.
This $n$-dimensional plane has a subset which is an $n-1$ dimensional plane described by all those points which have $\lambda_{0} = 0$.
Now, this sub-plane divides the plane into two parts, one which has $\lambda_{0}>0$ and other which have $\lambda_{0}<0$.
Now consider two points, $w$ and $w'$ in one part, say for which $\lambda_{0}>0$. Consider the convex hull of the points $H =$ convexhull$(w,v_{1},v_{2},\ldots,v_{n})$ and $H' = $ convexhull$(w',v_{1},v_{2},\ldots,v_{n})$. It is quite intuitive that there are points in the intersection of the two hulls for which $\lambda_{0}>0$. But I'm unable to prove this fact.
Let us work entirely in the affine space spanned by the $v_k$.
Consider the barycentre $b=\frac1n\sum_{k=1}^n v_k$ of the simplex with vertices $v_1,\ldots,v_n$. Consider the cone $C$ with vertex $w$ spanned by the rays $wv_1,\ldots,v_0v_n$. Then $H=C\cap F$ where $F$ is defined by $\lambda_0\ge0$. Moreover $b$ lies in the interior of $C$. Similarly define $C'$. Then $b$ is in the interior of $C\cap C'$. Take a small open ball with centre $b$ contained in the open set $C\cap C'$. Some of this open ball satisfies $\lambda_0>0$ and so lies in $H\cap H'$.