Length-convexity and domination for Hitchin representations

Let $S$ be a closed oriented surface of genus $g\ge 2$. For any $n\ge 3$, let ${\text{Hit}}_n(S)$ be the Hitchin component and let ${\text{Hit}}_n(S{)}^{\mathrm{\prime }}$ be the subset of representations that take values in ${\mathrm{P}\mathrm{S}\mathrm{p}}_{2m}(\mathbb{R})$ if $n=2m$ and $\mathrm{P}\mathrm{S}\mathrm{O}(m,m+1)$ if $n=2m+1$. In this note we prove that for any Hitchin representation $\rho \in {\text{Hit}}_n(S)$ there is $j\in {\text{Hit}}_n(S{)}^{\mathrm{\prime }}$ such that $\rho$ dominates $j$ in the Hilbert length spectrum; this confirms a conjecture of Lee-Zhang and, independently, Tholozan who had proved this for $n=3$. The key step is establishing a convexity of the Hilbert length function in the Bonahon-Dreyer parametrization of ${\text{Hit}}_n(S)$; this uses a result from the recent work of Barman-Gupta. We also prove these convexity and domination results for translation length functions and their spectra.