A Carousel Property for Compact Convex Sets
Yiming Song

TL;DR
This paper establishes a new geometric property relating compact convex sets within polygons, generalizing previous results and demonstrating the sharpness of bounds for even-sided polygons.
Contribution
It introduces a novel carousel property for convex sets in polygons, extending prior theorems and providing sharp bounds for even-sided polygons.
Findings
Proves a new property for convex sets in polygons with more than the number of common supporting lines.
Generalizes previous results by Adaricheva--Bolat and Czédli.
Shows the bound is sharp for polygons with an even number of sides.
Abstract
We prove that if and are compact convex sets contained in a convex -gon with vertices , and is strictly greater than the number of common supporting lines of and , then there exist and such that is in the convex hull of and . This recovers and generalizes previous results of Adaricheva--Bolat and Cz{\'e}dli. We also show that this bound is sharp for even .
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Taxonomy
TopicsLimits and Structures in Graph Theory · Point processes and geometric inequalities · Computational Geometry and Mesh Generation
