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Visibility-monotonic polygon deflation. (English) Zbl 1322.52003
Summary: A deflated polygon is a polygon with no visibility crossings. We answer a question posed by S. L. Devadoss et al. [Contrib. Discrete Math. 7, No. 1, 68–81 (2012; Zbl 1317.52018)] by presenting a polygon that cannot be deformed via continuous visibility-decreasing motion into a deflated polygon. We show that the least $$n$$ for which there exists such an $$n$$-gon is seven. In order to demonstrate non-deflatability, we use a new combinatorial structure for polygons, the directed dual, which encodes the visibility properties of deflated polygons. We also show that any two deflated polygons with the same directed dual can be deformed, one into the other, through a visibility-preserving deformation.
##### MSC:
 52A10 Convex sets in $$2$$ dimensions (including convex curves) 52C25 Rigidity and flexibility of structures (aspects of discrete geometry)
##### Keywords:
polygons; reconfiguration; deflation
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