论文标题
凸体混合体积之间的不平等：Minkowski和的体积界限
Inequalities between mixed volumes of convex bodies: volume bounds for the Minkowski sum
论文作者
论文摘要
在对在激进分子中可求解的通用稀疏多项式系统进行分类的过程中，埃斯特罗夫最近表明，minkowski sum $ p_1+p_1+\ dots+p_d $ of $ d $ d $ dipermensional lattice polattice polytopes从上面的函数$ o（m^d^d^d iS $ m m offorce of the of上面$（p_1，\ dots，p_d）$。这是众所周知的Aleksandrov-Fenchel不平等的结果。埃斯特罗夫（Esterov）还提出了确定更尖锐的界限的问题。我们展示了如何使用混合量之间的其他关系来改善$ O（m^d）$的界限，这在渐近上是鲜明的。此外，我们证明了尺寸2和3中的尖锐的上限。我们的结果概括为具有至少一个体积的任意凸体的元素。
In the course of classifying generic sparse polynomial systems which are solvable in radicals, Esterov recently showed that the volume of the Minkowski sum $P_1+\dots+P_d$ of $d$-dimensional lattice polytopes is bounded from above by a function of order $O(m^{2^d})$, where $m$ is the mixed volume of the tuple $(P_1,\dots,P_d)$. This is a consequence of the well-known Aleksandrov-Fenchel inequality. Esterov also posed the problem of determining a sharper bound. We show how additional relations between mixed volumes can be employed to improve the bound to $O(m^d)$, which is asymptotically sharp. We furthermore prove a sharp exact upper bound in dimensions 2 and 3. Our results generalize to tuples of arbitrary convex bodies with volume at least one.
