Should the set $\{ u \in C^\infty (M) : \int_M f e^u d\mu > 0 \}$ be convex?

Question

Let $M$ be a smooth compact manifold and $f \in C^\infty (M)$ be a sign-changing function. I'm interested in the following question: Should the subset

$$\Big\{ u \in C^\infty (M) : \int_M f e^u d\mu > 0 \Big\} $$

of $C^\infty (M)$ be convex? Even when $M$ is an interval in $\mathbb R$, the answer could be useful.

Thank you in advance.