Whether the set of a symmetric copositive matrices $x^T A x \geq 0$ for $\forall x \in \mathbb{R}^n$ $x \geq 0$ is convex

Question

Whether the set of a $n \times n$ symmetric copositive matrices $x^T A x \geq 0$ $\forall x \in \mathbb{R}^n$ and $x \geq 0$ is convex?

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Attempt:

If I show that the set of the positive semidefinite (psd) matrices is convex, then the intersection of the set of psd matrices and $x \geq 0$ is also convex. Is that correct?

Answer 1

No need to do that! Note that if $A,B$ are copositive, then for any $0<\lambda<1$ and $x\ge 0$:$$x^T\Bigg(\lambda A+(1-\lambda)B\Bigg)x=\lambda x^TAx+(1-\lambda )x^TBx\ge 0$$which implies that also $\lambda A+(1-\lambda)B$ is copositive and hence the set of copositive matrices is convex.