The Ptolemy and Zbăganu constants of normed spaces

In every inner product space HH the Ptolemy inequality holds: the product of the diagonals of a quadrilateral is less than or equal to the sum of the products of the opposite sides. In other words, ‖x−y‖‖z−w‖≤‖x−z‖‖y−w‖+‖z−y‖‖x−w‖‖x−y‖‖z−w‖≤‖x−z‖‖y−w‖+‖z−y‖‖x−w‖ for any points w,x,y,zw,x,y,z in HH. It is known that for each normed space (X,‖⋅‖)(X,‖⋅‖), there exists a constant CC such that for any w,x,y,z∈Xw,x,y,z∈X, we have ‖x−y‖‖z−w‖≤C(‖x−z‖‖y−w‖+‖z−y‖‖x−w‖)‖x−y‖‖z−w‖≤C(‖x−z‖‖y−w‖+‖z−y‖‖x−w‖). The smallest such CC is called the Ptolemy constant of XX and is denoted by CP(X)CP(X). We study the relationships between this constant and the geometry of the space XX, and hence with metric fixed point theory. In particular, we relate the Ptolemy constant CPCP to the Zbăganu constant CZCZ, and prove that if XX is a Banach space with CZ(X)<(1+3)/2, then XX has (uniform) normal structure and therefore the fixed point property for nonexpansive mappings. We derive general lower and upper bounds for both CPCP and CZCZ, and calculate the precise values of these two constants for several normed spaces. We also present a number of conjectures and open problems.