Let $X$ be a compact manifold with boundary $\partial X = X_0 \cup X_1$, and let $\omega$ be a volume form on $X$. Suppose $f:X \rightarrow [0,\infty)$ is a smooth non-negative function.
Is it always true that there exists a path $c:[0,1] \rightarrow X$ with $c(0) \in X_0$ and $c(1) \in X_1$ and $c(t) \in X-\partial X$ for all $0<t<1$ with the property that
$$\int_X f \omega \ge \int_X \omega \cdot \int_0^1 f(c(t))dt?$$
This seems very plausible, yet I don't know how to try proving it. Could somebody help me?