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 (n≥3) with x₁<⋯<xₙ. Then P₁P₂̄,⋯ PₙP₁̄ form a convex n-gon that lies in the half-space {equation*}{ {H}=\{(x,y)| x and y≤ y_1+(x-x_1/x_n-x_1)(y_n-y_1)\}⊆{R²} } {equation*} if and only if the following inequality holds {equation} {y_i-yᵢ₋₁}{x_i-xᵢ₋₁} ≤ {yᵢ₊₁-yᵢ}{xᵢ₊₁-xᵢ} 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.