On the local moduli of squareness
We introduce the notions of pointwise modulus of squareness and local modulus of squareness of a normed space X. This answers a question of C. Benítez, K. Przesławski and D. Yost about the definition of a sensible localization of the modulus of squareness. Geometrical properties of the norm of X (Fréchet smoothness, Gâteaux smoothness, local uniform convexity or strict convexity) are characterized in terms of the behaviour of these moduli.