Extremal structure of the set of absolute norms on R2 and the von Neumann–Jordan constant

The set of all absolute normalized norms on R2 (denoted by AN2) and the set of all convex functions ψ on [0,1] satisfying max{1−t,t}⩽ψ(t)⩽1 for t∈[0,1] (denoted by Ψ2) are identified by a one to one correspondence ψ(t)=‖(1−t,t)‖ψ for t∈[0,1]. The set AN2 has a convex structure which is isomorphic to that of Ψ2. In this paper, we determine the set of all extreme points of AN2 by considering the set Ψ2. Moreover, we will give the von Neumann–Jordan constant of (R2,‖⋅‖) when ‖⋅‖ is an extreme point of AN2.