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We show that a cone admits a gauge-reversing map if and only if it is asymmetric cone. We use this to prove that every isometry of a Hilbert geometryis a collineation unless the Hilbert geometry is the projective space of anon-Lorentzian symmetric cone, in which case the collineation group is of indextwo in the isometry group. We also determine the isometry group of the Thompsongeometry on a cone.

Gauge-reversing maps on cones, and Hilbert and Thompson isometries