On the entropy of Hilbert geometries of low regularities (Q6093594)

The aim of this paper is to prove two main results on the volume entropy of Hilbert geometries. The first one states that if \(\Omega\) is a convex and relatively compact domain of \({\mathbb R}^2\) which is Ahlfors \(\alpha\)-regular, then \(h(\Omega) = \frac{2\alpha}{\alpha +1}\), where \(h(\Omega)\) stands for the volume growth entropy. The second results strengthens the first main theorem of \textit{G. Berck} et al. [Pac. J. Math. 245, No. 2, 201--225 (2010; Zbl 1204.52003)] by weakening the assumption of \(C^{1,1}\)-regularity of the boundary of the convex set \(\Omega\).