Convexity in probabilistic Normed space

Abstract

The local convexity of a PN space is discussed using the idea of probability metric, and the constraint condition of convexity preservation of probabilistic normed space is given. Based on the probability background, it is proved that the probability norm and its linear combination are convex functions. When α ≥ −1, the lower horizontal set is convex. When α ≥ − ½, the lower level set can form a probabilistic normed space.