Given finitely many points $x_{1},\dots,x_{p} \in \mathbb{R}$, let $z$ be a convex combination of $\{x_{j}\}_{j=1}^{p}$, namely, suppose $\{\lambda_{j}\}_{j=1}^{p}$ are positive numbers satisfying $$\sum_{j=1}^{p}\lambda_{j}=1$$ then $$\sum_{j=1}^{p}\lambda_{j}x_{j}$$ is a convex combination of $\{x_{j}\}_{j=1}^{p}$.
If there are both positive and negative numbers in $\{x_{j}\}_{j=1}^{p}$,then there exists a convex combination that equals to zero. My question is how many such solutions there are that enable $\sum_{j=1}^{p}\lambda_{j}x_{j} = 0$?
Is if infinite? Is there a general conclusion for this?
You have $2$ equation with $p$ unknown on a simplex, hence the solution space is a simplex of dimension $p-2$.