Let $(\lambda\_1 , \cdots , \lambda\_d) \vdash k$ be a partition of $k$ of length $d$. Is there any way to decide if $0 \in \text{Conv}\{(\underbrace{\alpha\_1, \cdots, \alpha\_1}\_{\lambda\_1}, \cdots , \underbrace{\alpha\_d , \cdots, \alpha\_d}\_{\lambda\_d}) \in \mathbb{Z}^k: \, (\alpha\_1, \cdots , \alpha\_d) \in \mathbb{Z}^d,\, \sum\_{i=1}^d \alpha\_i = 0\}$, where not all $\alpha\_i$s are zero?



How to know if convex-hull of a set contains zero?
Like Emil noted in the comments, the question is equivalent to whether $0\in\mathrm{Conv}(\{\alpha\in\mathbb{Z}^d: \sum\_{i=1}^{d}\alpha\_i=0\}\setminus\{0\})$. To see that it is, note that it is the average of $(d-1,-1,-1,\ldots,-1)$ and its permutations.


