Isometries of polyhedral Hilbert geometries
نویسندگان
چکیده
We show that the isometry group of a polyhedral Hilbert geometry coincides with its group of collineations (projectivities) if and only if the polyhedron is not an n-simplex with n ≥ 2. Moreover, we determine the isometry group of the Hilbert geometry on the n-simplex for all n ≥ 2, and find that it has the collineation group as an index-two subgroup. These results confirm, for the class of polyhedral Hilbert geometries, several conjectures posed by P. de la Harpe. AMS Classification (2000): 53C60, 22F50
منابع مشابه
On the dynamics of isometries
We provide an analysis of the dynamics of isometries and semicontractions of metric spaces. Certain subsets of the boundary at infinity play a fundamental role and are identified completely for the standard boundaries of CAT(0)– spaces, Gromov hyperbolic spaces, Hilbert geometries, certain pseudoconvex domains, and partially for Thurston’s boundary of Teichmüller spaces. We present several rath...
متن کاملIsometries on Extremely Non-complex Banach Spaces
We construct an example of a real Banach space whose group of surjective isometries reduces to ± Id, but the group of surjective isometries of its dual contains the group of isometries of a separable infinite-dimensional Hilbert space as a subgroup. To do so, we present examples of extremely non-complex Banach spaces (i.e. spaces X such that ‖ Id+T ‖ = 1+‖T ‖ for every bounded linear operator T...
متن کاملEla Thompson Isometries on Positive Operators: the 2-dimensional Case∗
In this paper, a former result of the first author is completed. The structure is determined of all surjective isometries of the space of invertible positive operators on the 2-dimensional Hilbert space equipped with the Thompson metric or the Hilbert projective metric.
متن کاملThe Geometric Phase and Ray Space Isometries
We study the behaviour of the geometric phase under isometries of the ray space. This leads to a better understanding of a theorem first proved byWigner: isometries of the ray space can always be realised as projections of unitary or anti-unitary transformations on the Hilbert space. We suggest that the construction involved in Wigner’s proof is best viewed as an use of the Pancharatnam connect...
متن کاملIsometries on Spaces of Vector Valued Lipschitz Functions
This paper gives a characterization of a class of surjective isometries on spaces of Lipschitz functions with values in a finite dimensional complex Hilbert space.
متن کامل