Exhibiting a deep connection between purely geometric problems and real algebra, the complexity class\documentclass[12pt]{minimal}
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				\begin{document}$$\exists \mathbb {R}$$\end{document}∃Rplays a crucial role in the study of geometric problems. Sometimes\documentclass[12pt]{minimal}
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				\begin{document}$$\exists \mathbb {R}$$\end{document}∃Ris referred to as the ‘real analog’ ofNP. WhileNPis a class of computational problems that deals with existentially quantifiedbooleanvariables,\documentclass[12pt]{minimal}
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				\begin{document}$$\exists \mathbb {R}$$\end{document}∃Rdeals with existentially quantifiedrealvariables. In analogy to\documentclass[12pt]{minimal}
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				\begin{document}$$\Pi _2^p$$\end{document}Π2pand\documentclass[12pt]{minimal}
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				\begin{document}$$\Sigma _2^p$$\end{document}Σ2pin the famous polynomial hierarchy, we study the complexity classes\documentclass[12pt]{minimal}
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				\begin{document}$$\forall \exists \mathbb {R}$$\end{document}∀∃Rand\documentclass[12pt]{minimal}
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				\begin{document}$$ \exists \forall \mathbb {R}$$\end{document}∃∀Rwithrealvariables. Our main interest is theAreaUniversalityproblem, where we are given a plane graphG, and ask if for each assignment of areas to the inner faces ofG, there exists a straight-line drawing ofGrealizing the assigned areas. We conjecture thatAreaUniversalityis\documentclass[12pt]{minimal}
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				\begin{document}$$\forall \exists \mathbb {R}$$\end{document}∀∃R-complete and support this conjecture by proving\documentclass[12pt]{minimal}
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				\begin{document}$$\exists \mathbb {R}$$\end{document}∃R- and\documentclass[12pt]{minimal}
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				\begin{document}$$\forall \exists \mathbb {R}$$\end{document}∀∃R-completeness of two variants ofAreaUniversality. To this end, we introduce tools to prove\documentclass[12pt]{minimal}
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				\begin{document}$$\forall \exists \mathbb {R}$$\end{document}∀∃R-hardness and membership. Finally, we present geometric problems as candidates for\documentclass[12pt]{minimal}
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				\begin{document}$$\forall \exists \mathbb {R}$$\end{document}∀∃R-complete problems. These problems have connections to the concepts of imprecision, robustness, and extendability.