Fix $k,n \in \mathbb{N}$ such that $0 < k < n$ and define $v_i := \begin{pmatrix} \sin(2 \pi \cdot \frac{i}{n}) \\ \cos(2 \pi \cdot \frac{i}{n}) \end{pmatrix}$ for $i = 1,...,n$, then all $v_i$ lie on the unit circle and they are the vertices of a regular convex $n$-Gon, in clockwise order.
We can assing a point to every subset $I \subseteq \{1, ..., n\}$ with cardinality $|I|=k$, by setting $w_I := \sum_{i \in I} v_i$. Consider the special $k$-subsets given by the cyclic intervals $I_j := [j, j+k-1]$ (taken modulo $n$ i.e. $I_1 = \{1, ... , k\}, I_2 = \{2, ..., k+1\}, ... , I_n = \{ n, 1, ..., k-1 \}$ ). It is not too difficult to see that the vectors $w_1, ... , w_n$ defined by $w_j := w_{I_j}$ are again the vertices of a convex $n$-Gon, which we will call $P$.
Plotting the points $w_I$ for small values of $k$ and $n$, one observes that for every $k$-subset $I \neq I_1, ... ,I_n$ the point $w_I$ lies strictly inside the polygon $P$.
Here is an example where $n = 7$ and $k=2$.
The vector $x_1 = v_1 + v_3$ as well as $x_2 = v_5 + v_1$ lie strictly inside the polygon $P$ formed by $w_1, ... ,w_7$.
$\textbf{Now my Question:}$ After fixing $k$ and $n$, do all vectors $w_I$ with $I \subseteq \{1,...,n\}$ and $|I|=k$ lie (strictly, if $I \neq I_j$ as above) inside of the convex $n$-Gon formed by $w_1, ..., w_n$ ?