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 form at most ⌊n/k⌋ regular k-gons for any integer k ≥ 4 and that this bound is optimal, and (iv) provide a short proof that the sum of all the distances between its vertices is at least (n− 1)/2 and at most ⌊n/2⌋⌈n/2⌉(1/2) as long as the convex n-gon has unit perimeter.