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

2025-06-23

AI Summary

This paper investigates the computational complexity of determining star-shapedness for nonempty compact smooth domains—i.e., whether there exists a “kernel point” such that the line segment between it and any point in the set lies entirely within the set. Using the Real RAM model and Krasnosel’skiĭ’s theorem from convex geometry, the authors establish, for the first time, that this decision problem is ∀ℝ-complete—refuting the prior intuition that it resides at the ∃∀ level of the real hierarchy. This result precisely locates star-shapedness detection within the real-number complexity landscape. Moreover, the work systematically characterizes the real complexity of several related geometric properties—including convexity and visibility—thereby advancing the interface between computational geometry and real-number complexity theory.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Search and Optimization: Non-convex Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics
• Security and Privacy: Large-scale security measurements

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 $forallmathbb{R}$-complete. Since the obvious definition of star-shapedness has logical form $existsforall$, 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.

Problem

• Determine star-shaped set complexity in convex geometry
• Prove star-shaped testing is ∀ℝ-complete
• Explore complexity classifications using convex geometry theorems

Innovation

• Testing star-shapedness is $forallmathbb{R}$-complete
• Uses Krasnosel'skiĭ's theorem from convex geometry
• Studies complexity classifications in real hierarchy