Characterizations of convexities of normed spaces by means of $g$-angles

Abstract

The notion of $g$-angle has been defined by the author~[8]. This notion appeared to be very useful in characterisations of inner product spaces~[8]. Here we use the same notion in characterizations of strictly convex spaces, uniformly convex spaces, locally uniformly convex spaces, and normed spaces that are uniformly convex in every direction. Some corollaries of this characterizations are described as well. For example, we show that the so called quasi-inner product spaces are uniformly convex spaces, what has not been noted earlier.