On the degree of weak convergence of a sequence of finite measures to the unit measure under convexity

This is a study of the degree of weak convergence under convexity of a sequence of finite measures μ_j on ^k, k 1, to the unit measure δ_x0. LetQ denote a convex and compact subset of ^k, let ƒ ε C^m(Q), m 0, satisfy a convexity condition and let μ be a finite measure on Q. Using standard moment methods, upper bounds and best upper bounds are obtained for ¦∝_Qƒdμ − ƒ(x₀)¦. They sometimes lead to sharp inequalities which are attained for particular μ and ƒ. These estimates are better than the corresponding ones found in the literature.