convex hull of union of positive definite matrices

Question

Is it true that any element of ${\rm co}\Big\{\bigcup_{x \in [a,b]} S(x) \Big\}$ is in $\mathbb{S}_{> 0}^n$ (cone of positive definite $n \times n$ matrices), given that $S(x) \in \mathbb{S}_{> 0}^n$ for all $x \in [a,b] = \{\lambda a + (1 - \lambda)b \mid\lambda \in [0,1],\ a,b \in \mathbb{R}^n \}$?

Answer 1

Yes, I think so.

$z^t M z > 0$ for positive definite

Convex hull...

$z^t(\alpha M + (1-\alpha)N)z = \alpha z^tMz + (1-\alpha)z^tNz > 0$