In this paper, we introduce the  probabilistic normed groups. Among other results, we investigate the continuityof inner automorphisms of a group and the continuity of left and right shifts in probabilistic group-norm. We also study midconvex functions defined  on probabilistic normed groups and  give  some results about locally boundedness of such  functions.

in this paper, we introduce the  probabilistic normed groups. among other results, we investigate the continuityof inner automorphisms of a group and the continuity of left and right shifts in probabilistic group-norm. we also study midconvex functions defined  on probabilistic normed groups and  give  some results about locally boundedness of such  functions.

K–convex, K–midconvex and (K,λ)–convex set–valued maps are considered. Several conditions implying the continuity of such maps are collected.

We introduce the notion of strongly t-convex set-valued maps and present some properties of it. In particular, a Bernstein–Doetsch and Sierpiński-type theorems for strongly midconvex set-valued maps, as well as a Kuhn-type result are obtained. A representation of strongly t-convex set-valued maps in inner product spaces and a characterization of inner product spaces involving this representatio...

In this paper, we introduce an inner product on abelian groups and, after investigating the basic properties of product, first show that each group is a torsion-free normed group. We give examples such and describe norms induced by products. Among other results, Hilbert groups, midconvex orthogonal subgroups are presented, Riesz representation theorem divisible proved.