It has been shown by Soprunov that the normalized mixed volume (minus one) of an n-tuple of n-dimensional lattice polytopes is a lower bound for the number of interior lattice points in the Minkowski sum of the polytopes. He defined n-tuples of mixed degree at most one to be exactly those for which this lower bound is attained with equality, and posed the problem of a classification of such tuples. We give a finiteness result regarding this problem in general dimension $n\ge 4$, showing that all but finitely many n-tuples of mixed degree at most one admit a common lattice projection onto the unimodular simplex ${\mathrm{\Delta }}_{n-1}$. Furthermore, we give a complete solution in dimension $n=3$. In the course of this we show that our finiteness result does not extend to dimension $n=3$, as we describe infinite families of triples of mixed degree one not admitting a common lattice projection onto the unimodular triangle ${\mathrm{\Delta }}_2$.

它已被一个的Soprunov该归一化的混合体积（减一）所示ñ的元组Ñ维点阵多面体是下限为内部晶格点的在多面体的Minkowski求和的数目。他将最多混合度的n个元组定义为恰好达到其下界相等的那些元组，并提出了对此类元组进行分类的问题。我们从广义上给出关于该问题的有限性结果$ñ\ge 4$结果表明，除了有限个数外，所有其他n个元组的混合度最多只有一个，可以在单模单形上接受一个公共的点阵投影。${\mathrm{\Delta }}_{ñ-\mathrm{1个}}$。此外，我们提供了完整的尺寸解决方案$ñ=3$。在此过程中，我们证明了有限性结果未扩展至尺寸$ñ=3$，因为我们描述了一个混合度的三元组的无穷三族，一个不容许在单模三角形上有共同的晶格投影 ${\mathrm{\Delta }}_2$。