A set is star-shaped if there is a point in the set that can see every other point in the set in the sense that the line-segment connecting the points lies within the set. We show that testing whether a non-empty compact smooth region is star-shaped is $\forall\mathbb{R}$-complete. Since the obvious definition of star-shapedness has logical form $\exists\forall$, this is a somewhat surprising result, based on Krasnosel'skiĭ's theorem from convex geometry; we study several related complexity classifications in the real hierarchy based on other results from convex geometry.

翻译：一个集合若存在一点，使得该点与集合内任意其他点之间的连线线段均完全位于该集合内，则该集合被称为星形集合。我们证明，检验一个非空紧致光滑区域是否为星形是 $\forall\mathbb{R}$-完全的。由于星形性的显式定义具有 $\exists\forall$ 的逻辑形式，这一结果基于凸几何中的Krasnosel'skiĭ定理，显得颇为意外；我们基于凸几何中的其他结果，进一步研究了实层次结构中若干相关的复杂性分类问题。