  In order to investigate minimal sufficient conditions for an abstract
integral to belong to the convex hull of the integrand, we propose a system of
axioms under which it happens. If the integrand is a continuous $R^n$-valued
function over a path connected topological space, we prove that any such
integral can be represented as a convex combination of values of the integrand
in at most $n$ points, which yields an ultimate multivariate mean value
theorem.
