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It is shown that among all plane Hilbert geometries, the hyperbolic plane has the maximal volume entropy. More precisely, it is shown that the volume entropy is bounded above by 2 3−d ≤ 1, where d is the Minkowski dimension of the extremal set of K. An explicit example of a plane Hilbert geometry with noninteger volume entropy is constructed. In arbitrary dimension, the hyperbolic space has maximal entropy among all Hilbert geometries satisfying some additional technical hypothesis. To achievedoi:10.2140/pjm.2010.245.201 fatcat:dtao6bci3rht5hjnd55omzurju