Convex Sequence and Convex Polygon

In this paper, we deal with the question; under what conditions the points Pᵢ (xi, yi) (i = 1, , n) form a convex polygon provided x₁ < < xₙ holds. One of the main findings of the paper can be stated as follows: "Let P₁ (x₁, y₁), , Pₙ (xₙ, yₙ) are n distinct points (n3) with x₁<<xₙ. Then P₁P₂, PₙP₁ form a convex n-gon that lies in the half-space equation* {H=\ (x, y) | x and y y₁+ (x-x₁{xₙ-x₁) (yₙ-y₁) \}R^{2} } equation* if and only if the following inequality holds equation yᵢ-y₈-₁xᵢ-x₈-₁ y₈+₁-ₘ_₈x₈+₁-x₈ for all i\2, , n-1\. " equation Based on this result, we establish a linkage between the property of sequential convexity and convex polygon. We show that in a plane if any n points are scattered in such a way that their horizontal and vertical distances preserve some specific monotonic properties; then those points form a 2-dimensional convex polytope.