Regularity and splitting of directed minimal cones

We show that directed minimal cones in (n+1)-dimensional Euclidean space which have at most one singularity are - besides the trivial cases: empty set, whole space - half spaces. Using blow-up techniques, this result can be used to get C^{1,lambda}-regularity for the measure-theoretic boundary of almost minimal Caccioppoli sets which are representable as subgraphs in R^n, n<=8. This provides a different method to obtain a result due to De Giorgi. We also prove a splitting theorem for general directed minimal cones. Such a cone is the Cartesian product of R^k and C, where C is an undirected minimal cone or a half-line.