Abstract: This talk is based on my joint work with my student, Laltanpuia (Mizoram University). Let μ be a non-zero, non-atomic vector measure on the measurable space (X, )taking values in Rn. Liapounovs(Izv. Akad. Nauk. SSSR, 1940, 1941) convexity theorem gives that the range R(μ) is compact and convex. An immediate consequence of this is that there exist uncountably many smaller subcollections S of ith preservation of the range, that is R(μ) = R(μ/S). In case X is a topological space and , the class of Borel sets, such reductions consisting of open sets or other sets related to continuous functions on X have been obtained by H.Render and H.Stroetmann (Arch. Math., 1996), D.Wulbert ([Proc. Amer. Math. Soc., 1990 &2000; Israel J.Math., 2000;, J. Functional Analysis, 2001) and H.G.Kellerer Arch.Math., 1999; Proc. Amer. Math. Soc., 2002). JoséM.Gouweleeuw (Indag.Math. ,1993; Proc. London Math. Soc.3, 1995) gave a decomposition of an Rn -valued vector measure μ = (μ1, μ2, . . . , μn) on the measurable space (X, ) where each μi is a non-negative real measure on (X, ) based on the atoms of the measure μ. She also characterized those μ which have a convex range. Jan van Mill and André Ran (Indag. Mathem., 1996), gave various interesting variants and generalizations of the Gouweleeuw decomposition and convexity results. The first purpose of this talk is to display and apply these decompositions and make attempts to shorten the domain o various minimal subsets of alled shortenings, which preserve the range of μ. We also make use of the Rényi criteria (North-Holland, Amsterdam, 1970). and the work on interval filling sequences , particularly by Z. Daróczy, A. Járai, I. Katái and T. Szabó (Acta Sci. Math.,1986 &1988; Arch. Math., 1990), Next we shall display various quantized and approximation versions of the Liapounovs convexity Theorem given by C.A. Akemann and J. Anderson (Mem. AMS, 1991). We shall also give our quantized and approximation versions of shortenings. Examples, some based on those given by Liapounov himself and by J. Diestel and J. J. Uhl Jr. (AMS, 1977), will be given to illustrate the results. Various interpretations to mixed strategies versus pure strategies by D. Blackwell (Proc. AMS1951) and fair division / cake-cutting for instance, by L. Dubins and E.Spanier (Amer. Math. Monthly , 1961) and approximations / estimations by J. Elton and T.P.Hill (PAMS, 1987) will be a motivating factor all through.