Relations between geometric convexity, doubling measures and property $\Gamma$

Abstract:

In this article it is shown that the three conditions on the norm $\left \|\cdot \right \|$ of a Banach space called “geometric convexity”, “balanced” and “doubling” in an earlier work by the authors related to eikonal equations are in fact all equivalent. Moreover, each of them is equivalent to a condition called “Property $\Gamma$” by Ganichev and Kalton. A fifth condition, that the second derivative of the function $t\mapsto \left \|x+ty\right \|$ is a doubling measure on $[-2,2]$ for suitable $x, y\in X,$ is also equivalent to the various other properties, and this formulation occupies a central place in the analysis.