Consider points $a,b,c$ (not on a line) and $x_1,...,x_n$ in $\Bbb{R}^2$. I am looking for a necessary and sufficient condition in terms of the geometric configuration of these points such that for any convex function $f : \Bbb{R}^2 \longrightarrow \Bbb{R}$, $\frac{f(x_1)+\cdots+f(x_n)}{n} \leq \frac{f(a)+f(b)+f(c)}{3}$. (As a guess, I think the points $x_i$ must be in the triangle determined by $a,b,c$). If you know a reference or an idea please let me know.

In another direction I would like to generalize this to $f :\Bbb{R}^n \longrightarrow \Bbb{R}$. Any suggestion would be helpful.

Consider points $a,b,c$ (not on a line) and $x_1,...,x_n$ in $\Bbb{R}^2$. I am looking for a necessary and sufficient condition in terms of the geometric configuration of these points such that for any convex function $f : \Bbb{R}^2 \longrightarrow \Bbb{R}$, $\frac{f(x_1)+\cdots+f(x_n)}{n} \leq \frac{f(a)+f(b)+f(c)}{3}$. (As a guess, I think the points $x_i$ must be in the triangle determined by $a,b,c$). If you know a reference or an idea please let me know.