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Let $F$ be the average of $f(x)$ for $m$ given vectors $x_1,\dots,x_m$, that is,$$ F=\frac{1}{m}\sum_{i=1}^m f(x_i). $$ Here $f(x)$ is a convex function from $\mathbb{R}^n$ to $\mathbb{R}$.

Can we always find a convex combination of $x_1,\dots,x_m$, denoted by $\bar{x}=\sum_{i=1}^m\lambda_ix_i$, such that $F=f(\bar{x})$?

The convex combination implies that $\lambda_i\ge0$ and $\sum_{i=1}^m\lambda_i=1$.

I am aware of Jensen's inequality$$ F=\frac{1}{m}\sum_{i=1}^m f(x_i) \ge f(\bar{x}), $$ for $\bar{x}=\frac{1}{m}\sum_{i=1}^m x_i$ which may or may not be useful in a proof. I also know that $f_{min} \le F \le f_{max}$, where $f_{min}=\min_if(x_i)$ and $f_{max}=\max_if(x_i)$. If there are some counterexamples for the general case, then under what conditions we could find such a convex combination? Is continuity of $f$ a necessary condition?

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The function $f$ has full domain and is thus automatically continuous (see Rockafeallar's Convex Analysis). So the result follows from the Intermediate Value Theorem (connect on a line segment $x_i$ and $x_j$, where these two take on the minimum and the maximum).

max_zorn
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    Thanks but I'm not quite familiar with convex analysis. From this post I can see that $f$ is already continuous. I could not find the relevant theorem in Rockafeallar's Convex Analysis book. Could you mention which theorem in the book should I look at (or give a formal reference)? Is $f$ differentiable? I could not find how the Mean Value Theorem can be used to prove the result. –  Jul 16 '18 at 06:00
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  • Theorem 10.1.
  • g(t) := f((1-t)x_i+tx_j)$.
    • – max_zorn Jul 16 '18 at 06:07
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    Welcome to Convex Analysis, the most beautiful subject on this planet! – max_zorn Jul 16 '18 at 06:58