which we shall call local uniform convexity. Geometrically this differs from uniform convexity in that it is required that one end point of the variable chord remain fixed. In section I we prove a general theorem on the product of locally uniformly convex Banach spaces and with the aid of this theorem we establish that the two notions are actually different. Section II is devoted to the investigation of the relationship between local uniform convexity and strong differentiability of the norm. In section III we investigate conditions for isomorphism of a Banach space with a locally uniformly convex space.