Consider $x\_1,\cdots,x\_n \in \mathbb{R}^d$, and the closed convex cone in $\mathbb{R}^n$ defined by
$$\mathcal{K}(\underline{x}):=\{(\varphi(x\_1),\cdots,\varphi(x\_n)):\varphi \textrm{ convex on }\mathbb{R}^d\}.$$
I am looking for a good/efficient characterization for this cone or its polar cone. The motivation for this problem is trying to solve the projection of an given point $y\in \mathbb{R}^n$ onto $\mathcal{K}(\underline{x})$ numerically. In dimension $1$, a straightforward characterization would be given by $(\theta\_1,\cdots,\theta\_n) \in \mathcal{K}(\underline{x})$ if and only if
$$\frac{\theta\_{i+1}-\theta\_{i}}{x\_{i+1}-x\_i}\leq \frac{\theta\_{i+2}-\theta\_{i+1}}{x\_{i+2}-x\_{i+1}},\forall i=1,\cdots n-2.$$
This means that in one dimension, the projection problem can be solved by a simple quadratic programming with $n-2$ constraints. 


Comments and suggestions are greatly appreciated.



characterization of a certain closed convex cone
The paper <http://arxiv.org/abs/1402.1561> might be of use in case $d = 2$.


Your set $\mathcal{K}(\underline x)$ is denoted as $\mathrm{Conv}(X)$, see (3). A characterization of this set can be found in Theorem 1.4 and Theorem 1.8 provides certain relaxations, if your points are on a regular grid (subset of $\mathbb{Z}^2$).


In the case that your points $x\_i$ are in general position, you will end up with roughly $O(n^2)$ linear inequality constraints, see <http://dx.doi.org/10.1007/s002110000235>; Theorem 2.


Maybe the results can be generalized to $d > 2$, but you will end up with many linear constraints.


