It is proved that orthogonal convexity defined by A. JimenezMelado and E. Llorens-F'uster implies the weak Banach-Saks property. Relations between orthogonal convexity and another geometric properties, such as nearly uniform smoothness and property ( P ) , are studied. Introduction. Orthogonal convexity has been introduced by A. Jimenez-Melado and E. Llorens-F'uster (see [3] and [4]) as a geome...