Convex functions: bounding the difference

Suppose you are given a convex function $f: R^d \rightarrow R$. Let us say you are given $x,x' \in R^d$ and there are $x_1, x_2, \ldots, x_n \in R^d$ such that

$$\sum_{i=1}^n (x_i - x') = x - x'.$$

Is it possible to bound $f(x) - f(x')$ in terms of $f(x_i) - f(x')$?

That is, a bound of the form

$$f(x) - f(x') \leq \sum_{i}^n \left( f(x_i) - f(x') \right) + \sum_i^n \epsilon(x_i,x),$$

where $\epsilon_i$ are some small error functions based on some property of $f$?

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Crossposted to MathOverflow. In the future, please wait some time before posting your question in multiple fora, and when you do, provide links to the other posts - as you can imagine, it would be frustrating for someone to put time into answering your question here, only to see hear from you that you'd already gotten the solution elsewhere. –  Zev Chonoles Dec 23 '12 at 3:21