This is a refinement of the question Minimal representation of element in convex hull of $v_1,...,v_n\in\mathbb{R}^n$
Let $C$ a convex polyhedron in $\mathbb{R}^n$ and $v_1,..,v_k$ its extreme points and $x\in C$. Is there always a unique minimal representation of $x$ with respect to $v_1,...,v_k$ in $||.||_2$ the euclidean norm in the sense that, there is a unique $t=(t_1,...,t_k)\in[0,1]^k$ with $$x = \sum_i t_iv_i$$ and $||t||_2^2$ minimal over $[0,1]^n$? Seen through the lense of optimization, this transforms to the problem with $n+1$ linear constraints $$\min_{t_1,...,t_k\in[0,1]^k}\sum_{i} t_i^2$$ $$x = \sum_i t_iv_i$$ $$\sum_i t_i=1$$
Since $x\in C$, the feasible set of the problem is not empty and, therefore, by standard theory of convex optimization, such a $t\in[0,1]^k$ exists. But can I ensure uniqueness? I should get a global minimum for the $t$, so either a second $t'$ would attain the same value or there is none. By a feeling, everything here looks like a parabola, so there should be no second $t'$ but I am stuck how to show it.
