  A fundamental result in metabolic pathway analysis states that every flux
mode can be decomposed into a sum of elementary modes. However, only a
decomposition without cancelations is biochemically meaningful, since a
reversible reaction cannot have different directions in the contributing
elementary modes. This essential requirement has been largely overlooked by the
metabolic pathway community.
  Indeed, every flux mode can be decomposed into elementary modes without
cancelations. The result is an immediate consequence of a theorem by
Rockafellar which states that every element of a linear subspace is a conformal
sum (a sum without cancelations) of elementary vectors (support-minimal
vectors). In this work, we extend the theorem, first to "subspace cones" and
then to general polyhedral cones and polyhedra. Thereby, we refine Minkowski's
and Carath\'eodory's theorems, two fundamental results in polyhedral geometry.
We note that, in general, elementary vectors need not be support-minimal, in
fact, they are conformally non-decomposable and form a unique minimal set of
conformal generators.
  Our treatment is mathematically rigorous, but suitable for systems
biologists, since we give self-contained proofs for our results and use
concepts motivated by metabolic pathway analysis. In particular, we study cones
defined by linear subspaces and nonnegativity conditions - like the flux cone -
and use them to analyze general polyhedral cones and polyhedra.
  Finally, we review applications of elementary vectors and conformal sums in
metabolic pathway analysis.
