Let $(\lambda_1 , \cdots , \lambda_d) \vdash k$ be a partition of $k$ of length $d$ and suppose $\alpha_1, \cdots , \alpha_d$ are integers such that $\sum_{i=1}^d \alpha_i = 0$. Is there any way to decide if $0 \in \text{Conv}(\lambda_i \alpha_i) \subset \mathbb{Z}^d$?