On unit distances in a convex polygon

Algorithm for finding the largest inscribed rectangle in polygon

In many industrial and non-industrial applications, it is necessary to identify the largest inscribed rectangle in a certain shape. The problem is studied for convex and non-convex polygons. Another criterion is the direction of the rectangle: axis aligned or general. In this paper a heuristic algorithm is presented for finding the largest axis aligned inscribed rectangle in a general polygon. ...

The Sum of Distances between Vertices of a Convex Polygon with Unit Perimeter

On Viviani’s Theorem and its Extensions

Viviani’s theorem states that the sum of distances from any point inside an equilateral triangle to its sides is constant. We consider extensions of the theorem and show that any convex polygon can be divided into parallel segments such that the sum of the distances of the points to the sides on each segment is constant. A polygon possesses the CVS property if the sum of the distances from any ...

Metric inequalities for polygons

Let A1, A2, . . . , An be the vertices of a polygon with unit perimeter, that is ∑n i=1 |AiAi+1| = 1. We derive various tight estimates on the minimum and maximum values of the sum of pairwise distances, and respectively sum of pairwise squared distances among its vertices. In most cases such estimates on these sums in the literature were known only for convex polygons. In the second part, we t...

On Isosceles Triangles and Related Problems in a Convex Polygon

Given any convex n-gon, in this article, we: (i) prove that its vertices can form at most n2/2 + Θ(n log n) isosceles trianges with two sides of unit length and show that this bound is optimal in the first order, (ii) conjecture that its vertices can form at most 3n2/4 + o(n2) isosceles triangles and prove this conjecture for a special group of convex n-gons, (iii) prove that its vertices can f...