We present a short and simple proof of the recent result of Yang and Wang [12]. Stimulated by their idea, two geometric parameters UXα(ε) and βX(ε), both related to Gao's modulus of U-convexity of a Banach space X, are introduced. Their properties and the relationships with normal structure are studied. Some existing results involving normal structure and fixed points for non-expansive mappings in Banach spaces are improved.