My problem is to characterise (or find useful information on) the cone $C$ of $N\times N$ matrices $M$ ($N\geq 1$) such that $$V^t M V\geq 0$$ for every vector $V $ with non-negative entries. Is this cone of matrices familiar to anyone?

Remark 1: $C$ clearly contains the convex cone $S$ of semi-definite positive matrices, and that of matrices $P$ with non-negative entries. Even though it looks too easy, I could not prove that $C$ is not the convex hull of $S$ and $P$.

Remark 2: My original problem is to characterise the dual cone of $C$, containing the covariances of random vectors of $\mathbb{R}^N$ with a.s. non-negative entries.