Polygon Convexity: Another O(n) Test

An n-gon is defined as a sequence P = (V0, . . . , Vn−1) of n points on the plane. An n-gon P is said to be convex if the boundary of the convex hull of the set {V0, . . . , Vn−1} of the vertices of P coincides with the union of the edges [V0, V1], . . . , [Vn−1, V0]; if at that no three vertices of P are collinear then P is called strictly convex. We prove that an n-gon P with n > 3 is strictly convex if and only if a cyclic shift of the sequence (α0, . . . , αn−1) ∈ [0, 2π)n of the angles between the x-axis and the vectors −−−→ V0V1, . . . , −−−−−→ Vn−1V0 is strictly monotone. A “non-strict” version of this result is also proved. 1. Definitions and results A polygon is defined in this paper as any finite sequence of points (or, interchangeably, vectors) on the Euclidean plane R; the same definition was used in [5–8]. Let here P := (V0, . . . , Vn−1) be a polygon, which is sequence of n points; such a polygon is also called an n-gon. The points V0, . . . , Vn−1 are called the vertices of P . The smallest value that one may allow for the integer n is 0, corresponding to a polygon with no vertices, that is, to the sequence () of length 0. The segments, or closed intervals, [Vi, Vi+1] := conv{Vi, Vi+1} for i ∈ {0, . . . , n− 1} are called the edges of polygon P , where Vn := V0. The symbol conv denotes, as usual, the convex hull [9, page 12]. Note that, if Vi = Vi+1, then the edge [Vi, Vi+1] is a singleton set. For any two points U and V in R, let [U, V ] := conv{U, V }, [U, V ) := [U, V ] \ {V }, and (U, V ) := [U, V ] \ {U, V }, so that (U, V ) = ri [U, V ], the relative interior of [U, V ]. Let us define the convex hull and dimension of polygon P as, respectively, the convex hull and dimension of the set of its vertices: convP := conv{V0, . . . , Vn−1} and dimP := Date: February 1, 2008; file convex-poly/test/mehlhorn/arxiv1a.tex. 2000 Mathematics Subject Classification. Primary 52C45, 51E12, 52A10; Secondary 52A37, 03D15, 11Y16.