What kind of finite set that belongs to $\mathbb{R}^n$, with $P$ points $(P\geq n+1)$ satisfy both of the following conditions:
- There exist a set of $n+1$ points in $S$ such that every point in $\operatorname{Conv}(S)$ can be represented as a convex combination of these $n+1$ points.
- There does not exist any $k < n+1$ points such that every point in $\operatorname{Conv}(S)$ can be represented as a convex combination of these $n+1$ points.
Does this question ask me to find a specific finite set or just one with some restrictions?
From Caratheodory's theorem, given a set $S$ in $\mathbb{R}^n$, for every point $x$ in $\operatorname{Conv}(S)$ there exists a convex combination of no more than $n+1$ points from $S$ . I think this theorem guarantees that a finite set with cardinality more than $n+1$ can satisfy these two conditions. Am I missing something here?