 Solution

For $u\ge 0$, let ${\mathcal{C}}_u$ be the collection of interval components of the superlevel set $\{r:g(r)\ge u\}$. Two times $s,t$ lie in the same member of ${\mathcal{C}}_u$ exactly when $u\le m_g(s,t)$. ThereforeFor arbitrary real ${\lambda }_1,\dots ,{\lambda }_n$, the Tonelli theorem now givesThus $(m_g(s_i,s_j){)}_{i,j}$ is a positive semidefinite matrix.