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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 $f$?

  • 1
    [Crossposted to MathOverflow](http://mathoverflow.net/questions/117062). 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.2012-12-23

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