  In the present work, we consider Zuckerberg's method for geometric
convex-hull proofs introduced in [Geometric proofs for convex hull defining
formulations, Operations Research Letters 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 [Extended formulations for convex hulls of some bilinear
functions, Discrete Optimization 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.
