Demazure polynomials (called key polynomials in the paper) are multivariate polynomials \(\kappa_{\alpha}(x)\) defined for \(x\in \mathbb{R}^n\) and \(\alpha\in \mathbb{Z}^n_{\geq 0}\). They are the characters of the Demazure module of the general lineral group. The paper gives the combinatorial construction of those polytopes.
The Newton polytope of \(\kappa_{\alpha}(x)\) is the convex hull of the \(\gamma\) such that the coefficient \(c_{\gamma}\) of \(x^{\gamma}\) is non-zero in \(\kappa_{\alpha}(x)\).
Let \((\alpha_1, \dots, \alpha_n)\in \mathbb{Z}^n_{\geq 0}\). For \(1\leq i < j \leq n\), let \(t_{i,j}(\alpha)\) be the composition obtained from \(\alpha\) by interchanging \(\alpha_i\) and \(\alpha_j\). and let \(m_{i,j}(\alpha) = \alpha + e_i - e_j\). For a vector \(\beta\in \mathbb{Z}^n_{\geq 0}\) define \(\beta \leq_{k} \alpha\) if \(\beta\) can be generated from \(\alpha\) by applying a sequence of moves \(t_{i,j}\) for \(\alpha_i < \alpha_j\) and \(m_{i,j}\) for \(\alpha_i < \alpha_j - 1\).
The authors prove the Monikal-Tokcan-Yong conjecture, that is a vector is an exponent vector of the Newton polytope of \(\kappa_{\alpha}\) if and only if \(\beta \leq_k \alpha\). As a consequence, they prove that the Newton polytope of \(\kappa_{\beta}\) is contained in the Newton polytope of \(\kappa_{\alpha}\) if and only if \(\beta\leq_k \alpha\).