How Hard is it to be a Star? Convex Geometry and the Real Hierarchy
Computational Geometry
2025-11-20 v2 Computational Complexity
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 -complete. Since the obvious definition of star-shapedness has logical form , this is a somewhat surprising result, based on Krasnosel'ski\u{\i}'s theorem from convex geometry; we study several related complexity classifications in the real hierarchy based on other results from convex geometry.
Keywords
Cite
@article{arxiv.2506.18818,
title = {How Hard is it to be a Star? Convex Geometry and the Real Hierarchy},
author = {Marcus Schaefer and Daniel Štefankovič},
journal= {arXiv preprint arXiv:2506.18818},
year = {2025}
}