Existence of a concave function passing through given points

Question

I am wondering where to find proof or reference for the following fact:

Given variable sequence $\{ x_t \}_{t \in [T]} \subseteq \mathbb{R}^d$ and function value sequence $\{ y_t \}_{t \in [T]} \subseteq \mathbb{R}$, when would it be possible to find a concave function $f$ such that $f(x_t) = y_t$?

I am aware that someone asked a question about the existence of a convex function that interpolates given gradients and values (https://mathoverflow.net/users/29697/usul), Existence of a strictly convex function interpolating given gradients and values, URL (version: 2019-04-13): https://mathoverflow.net/q/327940.

I originally thought my question is similar and trivial but realized that it actually needs some thinking. It's kind of like the machine learning problem, where we have some inputs and outputs and we want to find some "mapping" between them. However, here the additional requirement is that the "mapping" must be a concave function. Does there always exist such a concave function? Or maybe we need some additional constraints on the given $\{ x_t \}_{t \in [T]} $ and $\{ y_t \}_{t \in [T]} $ in order to guarantee the existence?

Answer 1

Conditions can be found in Theorem 2 and Theorem 3 of

G. Carlier, T. Lachand-Robert, and B. Maury. A numerical approach to variational problems subject to convexity constraint. Numerische Mathematik, 2001.