In the study of geometric problems, the complexity class ƎR plays a crucial role since it exhibits a deep connection between purely geometric problems and real algebra. Sometimes ƎR is referred to as the "real analogue" to the class NP. While NP is a class of computational problems that deals with existentially quantified boolean variables, ƎR deals with existentially quantified real variables. In analogy to ∏_2~p and Σ_2~p in the famous polynomial hierarchy, we study the complexity classes ∀ƎR and Э∀R with real variables. Our main interest is focused on the Area Universality problem, where we are given a plane graph G, and ask if for each assignment of areas to the inner faces of G there is an area-realizing straight-line drawing of G. We conjecture that the problem Area Universality is ∀ƎR-complete and support this conjecture by a series of partial results, where we prove ƎR-and ∀ƎR-completeness of variants of Area Universality. To do so, we also introduce first tools to study ∀ƎR. Finally, we present geometric problems as candidates for ∀ƎR-complete problems. These problems have connections to the concepts of imprecision, robustness, and extendability.
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