Abstract: In this article, a new hypothesis on facial beauty perception is proposed: the weighted average of two facial geometric features is more attractive than the inferior one between them. Extensive evidences support the new hypothesis. We collected 390 well-known beautiful face images (e.g., Miss Universe, movie stars, and super models) as well as 409 common face images from multiple sources. Dozens of volunteers rated the face images according to their attractiveness. Statistical regression models are trained on this database. Under the empirical risk principle, the hypothesis is tested on 318,801 pairs of images and receives consistently supportive results. A corollary of the hypothesis is attractive facial geometric features construct a convex set. This corollary derives a convex hull based face beautification method, which guarantees attractiveness and minimizes the before--after difference. Experimental results show its superiority to state-of-the-art geometric based face beautification methods. Moreover, the mainstream hypotheses on facial beauty perception (e.g., the averageness, symmetry, and golden ratio hypotheses) are proved to be compatible with the proposed hypothesis.