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Let $\Omega$ be a strictly convex divisible subset of the $n$-dimensionalreal projective space which is not an ellipsoid. Even though $\partial\Omega$is not $C^2$, Benoist showed that it is $C^{1+\alpha}$ for some $\alpha>0$, andCrampon established that $\partial\Omega$ actually possesses a sort ofanisotropic H\"older regularity -- described by a list$\alpha_1\leq\dots\leq\alpha_{n-1}$ of positive real numbers -- at almost allof its points. In this article, we show that $\partial\Omega$ is maximallyanisotropic in the sense that this list of approximate regularities of$\partial\Omega$ does not contain repetitions. This result is a consequence ofthe simplicity of the Lyapunov spectrum of the Hilbert geodesic flow for everyequilibrium measure associated to a H\"older potential.

Simplicity of Lyapunov spectra and boundaries of non-conical strictly convex divisible sets