$$\forall \exists \mathbb {R}$$-Completeness and Area-Universality

Abstract

In the study of geometric problems, the complexity class \(\exists \mathbb {R}\) plays a crucial role since it exhibits a deep connection between purely geometric problems and real algebra. Sometimes \(\exists \mathbb {R}\) 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, \(\exists \mathbb {R}\) deals with existentially quantified real variables.