Convex Sequence and Convex Polygon

In this paper, we deal with the question: under what conditions n distinct points P_i(x_i, y_i) (i = 1,…, n) provided x₁ <…< x_n form a convex polygon? One of the main findings of the paper can be stated as follows: Let P₁(x₁, …, y₁),…, P_n(x_n, y_n) be n distinct points (n ≥ 3) with x₁ <…< x_n. Then P1P2¯,…,PnP1¯ \overline {{P_1}{P_2}} , \ldots ,\overline {{P_n}{P_1}} form a convex n-gon lying in the half-space ℋ¯=x,y|x∈ℝ and y≤y1+x−x1xn−x1(yn−y1)⊆ℝ2 \underline {\cal H} = \left\{{\left({x,y} \right)|x \in {\mathbb R}\,{\rm{and}}\,y \le {y_1} + \left({{{x - {x_1}} \over {{x_n} - {x_1}}}} \right)({y_n} - {y_1})} \right\} \subseteq {\mathbb R}^2 if and only if the following inequality holds yi−yi−1xi−xi−1≤yi+1−yixi+1−xifor alli∈{2,…,n−1}. \matrix{{{{{y_i} - {y_{i - 1}}} \over {{x_i} - {x_{i - 1}}}} \le {{{y_{i + 1}} - {y_i}} \over {{x_{i + 1}} - {x_i}}}} & {{\rm{for}}\,{\rm{all}}} & {i \in \{2, \ldots ,n - 1\}.} \cr} Based on this result, we establish a connection 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.