Extremals of the cone of norms and the cohomology of c3 -stratifications

Given real m x n-matrixces X and Y, when does there exist a stochastic matrix S such that SX = Y? The existence of S is characterized by the validity of an inequality which must hold for all positively homogeneous, convex, piecewise linear functions (so-called ph cpl functions) or, equivalently, for all extremals in the cone of all such functions. Every ph cpl function induces canonically a fan, that is a stratification of R^n by closed convex cones. We develop a cohomology theory for fans which characterizes these stratifications and also those which belong to extremals in the cone of all ph cpl functions. A centersymmetric version of the cohomology theory answers the same questions for the cone of piecewise linear seminorms. For n = 2, our theory implies a geometric interpretation of earlier work by Hardy, Littlewood, and Polya, as well as by Ruch, Schranner, and Seligman, based on the fact that in case n = 2 a ph cpl function is extremal only if it is the supremum of at most three linear functions. In case n >= 3, the situation changes drastically: For every k, the supremum of k linear functions is almost always an extremal ph cpl function. As another application of our theory we discuss a counterexample to a conjecture by Kakutani, constructed by A. Horn, and show how our theory could have been used to build large classes of such counterexamples.