منابع مشابه
And the Funk Metric
We discuss general notions of metrics and of Finsler structures which we call weak metrics and weak Finsler structures. Any convex domain carries a canonical weak Finsler structure, which we call its tautological weak Finsler structure. We compute distances in the tautological weak Finsler structure of a domain and we show that these are given by the so-called Funk weak metric. We conclude the ...
متن کاملFunk Metrics and R-Flat Sprays ∗
The well-known Funk metric F (x, y) is projectively flat with constant flag curvature K = −1/4 and the Hilbert metric Fh(x, y) := (F (x, y) + F (x,−y))/2 is projectively flat with constant curvature K = −1. These metrics are the special solutions to Hilbert’s Fourth Problem. In this paper, we construct a non-trivial R-flat spray using the Funk metric. It is then an inverse problem in the calcul...
متن کاملRemarks on the general Funk transform and thermoacoustic tomography
We discuss properties of a generalized Minkowski-Funk transform de ned for a family of hypersurfaces. We prove two-side estimates for the integral operator and show that the range conditions can be written in terms of the reciprocal Funk transform. Some applications to the spherical mean transform are considered. 1 Families of hypersurfaces Let X and be smooth manifolds of dimension n > 1 and F...
متن کاملGeneralized Minkowski-funk Transforms and Small Denominators on the Sphere
The Cauchy problem for the Euler-Poisson-Darboux equation on the unit sphere Sn gives rise to a family of fractional integrals M cos f(x) which integrate f over the spherical cap of radius centered at the point x 2 Sn. These fractional integrals are called the generalized Minkowski-Funk transforms because various transforms of integral geometry (including those of Minkowski and Funk) are partic...
متن کاملFourier transforms and the Funk–Hecke theorem in convex geometry
We apply Fourier transforms to homogeneous extensions of functions on Sn−1. This results in complex integral operators. The real and imaginary parts of these operators provide a pairing of stereological data that leads to new results concerning the determination of convex bodies as well as new settings for known results. Applying the Funk–Hecke theorem to these operators yields stability versio...
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ژورنال
عنوان ژورنال: Nature
سال: 1975
ISSN: 0028-0836,1476-4687
DOI: 10.1038/256529a0