Convexity properties of Yoshikawa–Sparr interpolation spaces

We study stability of the three geometric properties: uniform convexity, nearly uniform convexity, and property $\left(\beta \right)$ under the Yoshikawa–Sparr interpolation method when the resulting interpolation space is considered with various equivalent norms. We give an example which shows that interpolation spaces obtained by the discrete and continuous versions of the method need not be isometric and present a method of transferring geometric properties from the discrete case to the continuous one.