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

The set of all absolute normalized norms on R 2 (denoted by AN 2 ) 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 Ψ 2 ) are identified by a one to one correspondence ψ ( t ) = ‖ ( 1 − t , t ) ‖ ψ for t ∈ [ 0 , 1 ] . The set AN 2 has a convex structure which is isomorphic to that of Ψ 2 . In this paper, we determine the set of all extreme points of AN 2 by considering the set Ψ 2 . Moreover, we will give the von Neumann–Jordan constant of ( R 2 , ‖ ⋅ ‖ ) when ‖ ⋅ ‖ is an extreme point of AN 2 .