Cooperative positioning/orientation control of mobile heterogeneous anisotropic sensor networks for area coverage
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. This paper is concerned with proving a couple of Helly-type theorems for translates and homothets of convex curves on the Euclidean plane. It turns out that these Helly-type properties are very much like that of a circle and, as was amazing enough for me, admit of relatively simple and elementary proofs, which I present here. Throughout this paper, a convex curve C is defined to be the boundary of a proper convex subset B of Euclidean plane R, i.e. C = ∂B. A strictly convex curve is a convex curve containing no line segment. Given C ⊂ R, we call any set of the form λC + v, where λ > 0 and v ∈ R its homothet and, specifically, its translate, if λ = 1. Classically (e.g., ), the Helly number of a family F of sets is the minimal number χ satisfying the following property: if in a subfamily F ⊂ F any subfamily F0 ⊂ F with ♯F0 ≤ χ has a nonempty intersection, then the whole F has nonempty intersection. Helly theorem states that the Helly number of the family of all closed convex sets on the plane is 3. The Helly number of the family of all plane circles is 4, which is known from high-school geometry. We define the translation order μTr of C to be the minimal number μ such that: for any subset S ⊂ R such that for any S0 ⊂ S with ♯S0 ≤ 0 points there is a translate C ′ (depending on S0) of C containing S0, then there exists a translate of C containing the whole set S Analogously the homothety order μHt of C is defined. ”Duals” of the facts above are that the translation order of every closed convex set on the plane is 3 and that of a circle is 4. Our aim is to prove the following three theorems. Theorem 1 (i) If C is a convex plane curve, then μTr(C), μHt(C) ≤ 6.