Characterizing circles by a convex combinatorial property

Abstract

Abstract. Let K 0 be a compact convex subset of the plane R 2 , and assume that K 1 ⊆ R 2 is similar to K 0 , that is, K 1 is the image of K 0 with respect to a similarity transformation R 2 → R 2 . Kira Adaricheva and Madina Bolat have recently proved that if K 0 is a disk and both K 0 and K 1 are included in a triangle with vertices A 0 , A 1 , and A 2 , then there exist a j ∈ {0, 1, 2} and a k ∈ {0, 1} such thatHere we prove that this property characterizes disks among compact convex subsets of the plane. In fact, 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. Aim and introductionOur goal. The real plane and the usual convex hull operator on it will be denoted by R 2 and Conv, respectively. That is, for a set X ⊆ R 2 of points, Conv(X) is the smallest convex subset of R 2 that contains X. As usual, if X and Y are subsets of R 2 such that Y = ϕ(X) for a similarity transformation or an isometric transformation ϕ : R 2 → R 2 , then X and Y are similar or isometric (also called congruent), respectively. Disks are convex hulls of circles and circles are boundaries of disks. The singleton subsets of R 2 are both disks and circles. A compact subset of R 2 is a topologically closed and bounded subset. Our aim is to prove the following theorem. Theorem 1.1. If K 0 is a compact convex subset of the plane R 2 , then the following three conditions are equivalent.(i) K 0 is a disk.(ii) For every K 1 ⊆ R 2 and for arbitrary points A 0 , A 1 , A 2 ∈ R 2 , if K 1 is similar to K 0 and both K 0 and K 1 are contained in the triangle Conv({A 0 , A 1 , A 2 }), then there exist a j ∈ {0, 1, 2} and a k ∈ {0, 1} such that(iii) The same as the second condition but "similar" is replaced by "isometric".Our main achievement is that (iii) implies (i). The implication (i) ⇒ (ii) was discovered and proved by Adaricheva and Bolat [3]; see also Czédli [9] for a shorter proof. The implication (ii) ⇒ (iii) is trivial.

Circles and crossing planar compact convex sets

“…First, we recall some notations, well-known concepts, and well-known facts from Czédli [14] and Czédli and Stachó [23]. In order to ease our terminology, let us agree that every convex set in this paper is assumed to be nonempty, even if this is not always mentioned.…”

“…Armed with Definition 2.1, we recall the following statement from Czédli [14]. Theorem 2.2 (Lemma 3.3 in [14]).…”

“…Armed with Definition 2.1, we recall the following statement from Czédli [14]. Theorem 2.2 (Lemma 3.3 in [14]). For every nonempty compact convex subset X of the Euclidean plane R 2 , the following two conditions are equivalent.…”

A convex combinatorial property of compact sets in the plane and its roots in lattice theory

“…An earlier attempt to generalize Adaricheva and Bolat [2, Theorem 3.1], see Corollary 1.2 here, did not use homotheties and resulted in a new characterization of disks. Namely, for a convex compact set U 0 ⊆ R 2 , Czédli [14] proved that U 0 is a disk if and only if for every isometric copy U 1 of U 0 and for any points…”

“…Since the rest of the paper focuses mainly on homotheties in our sense, we use disjunction rather than the hyphened form "homothety-translation". Second, it is easy to see that (1.2) does not hold for two arbitrary compact sets, so the disjunction of (a), (b), and (c) cannot be omitted from Theorem 1.1; see also Czédli [14] for related information. Third, Example 4.1 of Czédli [15] rules out the possibility of generalizing Theorem 1.1 for higher dimensions.…”