How Hard is it to be a Star? Convex Geometry and the Real Hierarchy

TL;DR

This work proves that testing star-shapedness for non-empty compact smooth semialgebraic regions is $\forall \mathbb{R}$-complete, by leveraging Krasnosel'skiĭ's duality between star-shapedness and common visibility of points. It presents a $orall \mathbb{R}$-membership framework for smooth regions based on tangent-half-space representations and gradient conditions, and develops a robust $\forall \mathbb{R}$-hardness reduction using effective Sard-type perturbations to encode polynomial-zero existence within a unit ball. The results illustrate how convex-geometry dualities can yield unexpected complexity reductions in the real hierarchy, complementing classical Carathéodory/Steinitz/Kirchberger theorems, and open multiple avenues for extending the approach to broader semialgebraic classes and related geometric properties.

Abstract

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.

Figures (2)

• Figure 1: A closed set $S$ in $\mathbb{R}^2$ in which every three points can see a common point, but $S$ is not star-shaped: every point in $S$ can see at most finitely many points on the $y$-axis. Example due to Hare and Kenelly HK68.
• Figure 2: Moving a ball $B$ along $xy$ until it touches the boundary of $S$ in the spherical point $z$. In the drawing we have $x \not\in S$, but $x \in S$ is allowed.

Theorems & Definitions (36)

• Theorem 1.1
• Remark 1.2: Assumptions
• Lemma 1.3: Fredholm
• Lemma 1.4: Farkas
• Theorem 2.1: Carathéodory
• Theorem 2.2
• proof
• Theorem 2.3: Steinitz
• Corollary 2.4
• proof : Proof of Corollary \ref{['cor:inintconvER']}