Consider a compact Kaehler manifold $(M, \omega).$ Fixing the Kaehler class $[\omega]$ we can parametrize the space of Kaehler metrics with respect to that Kaehler class:
$$K:= \{ \varphi \in C^{\infty}(M,\mathbb{R}) | \omega + i \partial \bar{\partial} \varphi >0 \} $$
I want to prove (if that is true) that $K$ is convex. To do so, consider $\psi, \varphi \in K,$ I have to show that $t \psi + (1-t) \varphi \in K$ for all $t \in [0,1].$ At the boundary of the interval there are no problems, namely for $t=0$ and for $t=1$ the claim follows since we have chosen $\varphi$ and $\psi$ in $K.$ What is left to show is that for $0<t<1$ that convex combination is still in $K.$ Any helps? Is there a way to use compactness?
Write
$$\omega + i\partial \bar\partial (t \psi + (1-t) \varphi ) = t (\omega + i\partial \bar \partial \psi) + (1-t) (\omega + i\partial \bar\partial \varphi)$$
and both terms are positive.