Consider a polynomial optimization problem of the following type
\begin{equation} \begin{array}{cl} \text{maximize} & p(\mathbf{x}) \\ \text{subject to} & \mathbf{x} \in K \\ \end{array} \end{equation}
where $p \in \mathbb{R} [x_1,x_2,\dots,x_n]$ is a multivariate polynomial and $K \subset \mathbb{R}^n$ is a semialgebraic set (i.e., a set defined by some polynomial inequalities and equalities).
Lasserre introduced a hierarchy of relaxations such that every relaxation has a dual (which is called sum of squares relaxation). If I'm not wrong, the duality gap between this primal and the dual relaxations is zero when $K$ has an interior point and this is a sufficient condition. I'm wondering if there is any other situation such that it makes the duality gap zero between this primal and dual?
For example, in the paper Strong duality in Lasserre's hierarchy for polynomial optimization, it is said that if $K$ is compact then there is no duality gap but they add new polynomial constraint to $K$ which I think change the Lasserre's relaxation of its dual from.