Characterizing circles by a convex combinatorial property

2016/11/28 by Czédli, Gábor
#52A01 (Secondary) #52C99 (Primary) #Combinatorics (math.CO) #FOS: Mathematics #Metric Geometry (math.MG)

paper · doi:10.48550/arxiv.1611.09331

Abstract

Let K₀ be a compact convex subset of the plane \mathbb R², and assume that K₁⊆ \mathbb R² is similar to K₀, that is, K₁ is the image of K₀ with respect to a similarity transformation \mathbb R²→\mathbb R². Kira Adaricheva and Madina Bolat have recently proved that if K₀ is a disk and both K₀ and K₁ are included in a triangle with vertices A₀, A₁, and A₂, then there exist a j∈ \0,1,2\ and a k∈\0,1\ such that K_1-k is included in the convex hull of K_k∪(\A₀,A₁, A₂\∖\A_j\). Here we prove that this property characterizes disks among compact convex subsets of the plane. Actually, we prove even more since we replace "similar" by "isometric" (also called "congruent"). Circles are the boundaries of disks, so our result also gives a characterization of circles.

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