Lengths of tours and permutations on a vertex set of a convex polygon Original Research Article

Let x0,x1,…,xn−1 be vertices of a convex n-gon in the plane (each internal angle may be equal to π), where, (x0,x1),(x1,x2),…,(xn−2,xn−1), and (xn−1,x0) are edges of the n-gon. Denote the length of the line segment xixj by d(i,j). Let σ be a permutation on {0,1,…,n−1}. Define a length of σ as S(σ)=∑i=0n−1d(i,σ(i)). Further, define σp as σp(i)=i+p mod n for all i∈{0,1,…,n−1}. This paper shows that S(σp) is a strictly concave and strictly increasing function for 1⩽p⩽⌊n/2⌋. It is also shown that σ⌈n/2⌉ and σ⌊n/2⌋ are longest permutations and σ1 and σn−1 are shortest permutations under some restriction.