The truncated K-Moment problem for closure of open sets

We solve the truncated K-moment problem when $K\subseteq R^n$ is the closure of a, not necessarily bounded, open set (which includes the important cases $K=R^n$ and $K=R^n_+$). That is, we completely characterize the interior of the convex cone of finite sequences that have a representing measure on $K\subseteq R^n$. It is in fact the domain of the Legendre-Fenchel transform associated with a certain convex function. And so in this context, detecting whether a sequence is in the interior of this cone reduces to solving a finite-dimensional convex optimization problem. This latter problem is related to maximum entropy methods for approximating an unknown density from knowing only finitely many of its moments. Interestingly, the proposed approach is essentially geometric and of independent interest, as it also addresses the abstract problem of characterizing the interior of a convex cone C which is the conical hull of a set continuously parametrized by a compact set $M\subset R^n$ or $M \subseteq S^{n-1}$, where $M$ is the closure of an open subset of $R^n$ (resp. $S^{n-1}$). As a by-product we also obtain a barrier function for the cone C.