Suppose you are given a convex function $f: R^d \rightarrow R$. Let us say you are given $x,x' \in R^d$ and $x_1, x_2, \ldots, x_n \in R^d$ 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 $f$? Is there a class of functions $f$ for which this will be valid with $\epsilon = 0$?