In the present work, we consider Zuckerberg’s method for geometric convex-hull proofs introduced in Zuckerberg (Oper Res Lett 44(5):625–629, 2016). It has only been scarcely adopted in the literature so far, despite the great flexibility in designing algorithmic proofs for the completeness of polyhedral descriptions that it offers. We suspect that this is partly due to the rather heavy algebraic framework its original statement entails. This is why we present a much more lightweight and accessible approach to Zuckerberg’s proof technique, building on ideas from Gupte et al. (Discrete Optim 36:100569, 2020). We introduce the concept of set characterizations to replace the set-theoretic expressions needed in the original version and to facilitate the construction of algorithmic proof schemes. Along with this, we develop several different strategies to conduct Zuckerberg-type convex-hull proofs. Very importantly, we also show that our concept allows for a significant extension of Zuckerberg’s proof technique. While the original method was only applicable to 0/1-polytopes, our extended framework allows to treat arbitrary polyhedra and even general convex sets. We demonstrate this increase in expressive power by characterizing the convex hull of Boolean and bilinear functions over polytopal domains. All results are illustrated with indicative examples to underline the practical usefulness and wide applicability of our framework.

在目前的工作中，我们考虑了 Zuckerberg 中引入的用于几何凸包证明的 Zuckerberg 方法（Oper Res Lett 44(5):625–629, 2016）。尽管在设计算法证明以实现多面体描述的完整性方面具有很大的灵活性，但到目前为止，它在文献中几乎没有被采用。我们怀疑这部分是由于其原始陈述所包含的相当繁重的代数框架。这就是为什么我们基于 Gupte 等人的想法为扎克伯格的证明技术提供了一种更轻量级和更易于使用的方法。（离散优化 36：100569，2020）。我们引入了集合表征的概念来替换原始版本中所需的集合论表达式，并促进算法证明方案的构建。与此同时，我们开发了几种不同的策略来进行扎克伯格类型的凸包证明。非常重要的是，我们还表明我们的概念允许对扎克伯格的证明技术进行重大扩展。虽然原始方法仅适用于 0/1 多面体，但我们的扩展框架允许处理任意多面体甚至一般凸集。我们通过表征多面域上布尔函数和双线性函数的凸包来证明这种表达能力的增加。所有结果都用指示性示例进行说明，以强调我们框架的实际实用性和广泛适用性。我们的扩展框架允许处理任意多面体甚至一般凸集。我们通过表征多面域上布尔函数和双线性函数的凸包来证明这种表达能力的增加。所有结果都用指示性示例进行说明，以强调我们框架的实际实用性和广泛适用性。我们的扩展框架允许处理任意多面体甚至一般凸集。我们通过表征多面域上布尔函数和双线性函数的凸包来证明这种表达能力的增加。所有结果都用指示性示例进行说明，以强调我们框架的实际实用性和广泛适用性。