On Some Properties of Functions on Convex Galaxies

In this paper, we define and study extensively a new type of external sets in , we call it "convex galaxies". We show that these convex external sets may be classified in some definite types. More precisely, we obtain the following :
(1)Let be a convex galaxy which is symmetric with respect to zero, then
(i) is an - galaxy (0) if and only if there exists an internal strictly increasing sequence of strictly positive real numbers with such that and , for all , where, is some limited real number such that .
(ii) is a non-linear galaxy if and only if there exists an internal strictly increasing sequence of strictly positive real numbers with such that is unlimited for all .
(2)Let be a convex galaxy which is symmetric with respect to zero, then
(i) is an - galaxy (0) iff there exists a real internal strictly increasing - function , such that , and for all limited , where is a positive real number.
(ii) is a non-linear galaxy if and only if there exists a real internal strictly increasing - function , such that and is positive unlimited, for all appreciable .