In this working paper we study some properties of a particular mapping in Rn related to an optimization problem with one equality constraint. We motivate the definition of the relevant mapping starting from a portfolio selection problem, in which we minimize the risk of an investment (the variance of its return) with one equality constraint given by a fixed level of the return itself. The vector of the optimal portfolio is given by a particular mapping of the vector of returns and this mapping is taken into consideration. All the properties of this mapping may of course be considered in the more general context of an optimization problem with one equality contraint, but some of them may be reasonably extended in the further general case of more equality contraints. Although it has not been investigated in this work, some results may have a relevant meaning in explaining the relation between the vector of expected returns and the optimal portfolio.
