Let \(X\) be a smooth projective variety over \(\mathbb{C}\) and \(T\) an algebraic torus acting on \(X\) with isolated fixed points. Denote by \(W\subset X\) the fixed point set of \(T\). A generic one parameter subgroup \(\rho:\mathbb{C}^*\to T\) has the same fixed point set as \(T\). The attraction sets \[ X_w= \{x\in X\mid \lim_{\lambda\to 0} \rho(\lambda).x=w\} \] are known to be isomorphic to affine spaces for each \(w\in W\). In the special case of a flag variety \(X=G/P\) these attraction sets are just the Schubert cells. For this reason, \(X_w\) is called a Schubert-type cell in the general case. The author proves that the decompositions of \(X\) into the following two kinds of strata coincide:
(1) The union of all orbits of \(T\) whose images under a momentum map are the same convex polyhedron.
(2) The intersection of Schubert-type cells which are obtained by permuting the generic one parameter subgroup \(\rho\).
In the special case of a Grassmann manifold, this was proved by Gelfand-Goresky-MacPherson-Serganova and by Gelfand-Serganova in the case of a general flag variety \(X=G/P\). Furthermore, the notation of \((W,Q)\)-matroids due to Gelfand-Serganova is generalised to projective varieties.