Entropy of Hilbert metrics and length spectrum of Hitchin representations in $\mathrm{PSL}(3,\mathbb{R})$

Nicolas Tholozan

We prove a sharp inequality between the Blaschke and Hilbert distance on a proper convex domain: for any two points $x$ and $y$, \[d^B(x,y) < d^H(x,y) +1.\] We obtain two interesti…