Extremals of the cone of norms and the cohomology of \(c^ 3\)- stratifications
The problem considered is the following: Given real \(m\times n\) matrices \(X\) and \(Y\), when does there exist a stochastic matrix \(S\) such that \(SX=Y\)? The answer is given in terms of a cohomology theory for fans which are stratifications of \(R^ n\) by closed convex cones. There are links to a classical result of Hardy, Littlewood and Pólya, who described when two probability distributions on a finite set can be mapped one onto the other by a bistochastic matrix. There is a section devoted to cones and stratifications by closed convex cones. A cohomology theory is developed, where the first cohomology group measures to what extent the above mentioned problem can be solved. The most satisfactory result is obtained when \(n=2\). Finally a counterexample due to \textit{A. Horn} [cf. \textit{S. Sherman}, Proc. Am. Math. Soc. 3, 511-513 (1952; Zbl 0048.250) and ibid 5, 998-999 (1954; Zbl 0058.011)] to a conjecture by S. Kakutani is explained in the light of the present theory.
scientific article; zbMATH DE number 56225