Let $C$ denote the class of all real-valued convex functions on $[0, 1]^2$. Fix $n \geq 2$ and points $x_1, \dots, x_n$ in $[0, 1]^2$. Let $S \subset R^n$ be defined by \begin{equation*} S := \left\{ (f(x_1), \dots, f(x_n) : f \in C \right\}. \end{equation*} I would like to associate, for each $\theta \in S$, a function $f_{\theta} \in C$ such that, for a universal constant $K$, \begin{equation*} \frac{1}{n} \sum_{i=1}^n (\theta_i - \alpha_i)^2 \leq K \int_{[0, 1]^2} (f_{\theta} (x) - f_{\alpha}(x))^2 dx \end{equation*} for all $\theta,\alpha \in S$.
This clearly may not be possible for arbitrary $x_1, \dots, x_n$. The question is whether such an association is possible for at least some choices of $x_1, \dots, x_n$. In particular, is it true if $x_1, \dots, x_n$ are chosen to be regular grid points in $[0, 1]^2$.