Let X be a complete normal algebraic variety over an algebraically closed field, with an action of a reductive group G. Let \(T\subset G\) be a maximal torus, and let \(X^ T=X_ 1\cup...\cup X_ r\) be the decomposition into connected components of \(X^ T\). The aim of the paper under review is to define a generalized moment function \(f: \{X_ 1,...,X_ r\}\to X(T)\otimes {\mathbb{R}},\) where X(T) is the character group of T, with the following property: the set \(X^ s_ T=\{x\in X| \quad 0\in Int conv(\{f(X_ i);X_ i\cap Tx\neq \emptyset \})\}\) is open, T-invariant and admits a geometric quotient \(X^ s_ T\to X^ s_ T/T\) in the category of algebraic varieties. The results obtained may be considered as a step toward the solution of the general problem of describing all open G-invariant subsets U of X for which the geometric quotient \(U\to U/G\) exists.