How Hard is it to be a Star? Convex Geometry and the Real Hierarchy
Marcus Schaefer, Daniel \v{S}tefankovi\v{c}
TL;DR
This paper investigates the computational complexity of determining star-shapedness in smooth regions, revealing it is a highly complex problem classified as -complete, which is surprising given its logical form.
Contribution
It establishes the -completeness of testing star-shapedness for smooth regions, connecting convex geometry theorems with computational complexity classifications.
Findings
Testing star-shapedness is -complete.
The problem's complexity is surprising given its logical form.
The paper explores related classifications in the real hierarchy.
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.
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