Extremal circular permutations with concave functions

Abstract

Let { a 1, a 2,…, a n } be a set of n distinct real numbers and let α 1, α 2,…, α n be a permutation of the numbers. We construct the permutation to maximise L f =∑ i=1 n f(| α i+1 − α i |), for any increasing concave function f, where we denote α n+1 ≡ α 1. The optimal permutation depends on the particular numbers { a 1, a 2,…, a n } and the function f, contrary to a postulate by Chao and Liang (European J. Combin. 13 (1992) 325).