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 $\epsilon = 0$?



Convex functions: bounding the difference
$$f(x) - f(x') \le \sum\_i \left(f(x\_i) - f(x')\right)$$ 
iff the function $g(x) = f(x + x') - f(x')$ is subadditive. 


