Helly-type Theorems for Plane Convex Curves
Alexander Getmanenko (Yaroslavl State University; PennState, University)
TL;DR
This paper explores Helly-type theorems for families of convex curves, extending classical results from circles to more general convex shapes, with some weaker conditions for arbitrary convex curves.
Contribution
It generalizes Helly-type theorems from circles to strictly convex curves and provides weaker results for arbitrary convex curves.
Findings
Helly-type properties hold for translates and homothets of strictly convex curves.
Weaker Helly-type results are established for arbitrary convex curves.
The results extend classical circle-based Helly theorems to broader convex shapes.
Abstract
Families of translates and homothets of strictly convex curves are proven to possess Helly-type properties generalizing those of a circle. Weaker results are shown for arbitrary convex curves.
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Taxonomy
TopicsAlgebraic Geometry and Number Theory · Commutative Algebra and Its Applications · Point processes and geometric inequalities