,
István Szalkai
In this paper, we deal with the question: under what conditions n distinct points Pi(xi,yi) (i=1,...,n) provided x1<...<xn form a convex polygon? One of the main findings of the paper can be stated as follows: Let P1(x1,y1),..., Pn(xn,yn) be n distinct points (n≥3) with x1<...<xn. Then \overline{P1P2},...,\overline{PnP1} form a convex n-gon lying in the half-space
\underline{H} = {(x,y)| x∈ℝ and y≤y1 + [(x-x1)/(xn-x1)](yn-y1)} ⊆ ℝ2
if and only if the following inequality holds
(yi-yi-1)/(xi-xi-1) ≤ (yi+1-yi)/(xi+1-xi) for all i∈{2,...,n−1}.
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.
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2024
Published: 2024-01-18
10.2478/amsil
English