Crofton formulas in projective Finsler spaces
Authors:
J. C. Álvarez Paiva and E. Fernandes
Journal:
Electron. Res. Announc. Amer. Math. Soc. 4 (1998), 91-100
MSC (1991):
Primary 53C65; Secondary 53C60
DOI:
https://doi.org/10.1090/S1079-6762-98-00053-5
Published electronically:
November 23, 1998
MathSciNet review:
1655987
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Abstract: We extend the classical Crofton formulas in Euclidean integral geometry to Finsler metrics on $\mathbb {R}^n$ whose geodesics are straight lines.
- J.C. Álvarez Paiva, Integral geometry on Finsler manifolds, preprint 1998.
- J.C. Álvarez Paiva, Anti-self-dual symplectic forms and integral geometry, preprint 1998.
- J. C. Alvarez, I. M. Gelfand, and M. Smirnov, Crofton densities, symplectic geometry and Hilbert’s fourth problem, The Arnold-Gelfand mathematical seminars, Birkhäuser Boston, Boston, MA, 1997, pp. 77–92. MR 1429885
- V. I. Arnol′d and A. B. Givental′, Symplectic geometry, Current problems in mathematics. Fundamental directions, Vol. 4, Itogi Nauki i Tekhniki, Akad. Nauk SSSR, Vsesoyuz. Inst. Nauchn. i Tekhn. Inform., Moscow, 1985, pp. 5–139, 291 (Russian). MR 842908
- Arthur L. Besse, Manifolds all of whose geodesics are closed, Ergebnisse der Mathematik und ihrer Grenzgebiete [Results in Mathematics and Related Areas], vol. 93, Springer-Verlag, Berlin-New York, 1978. With appendices by D. B. A. Epstein, J.-P. Bourguignon, L. Bérard-Bergery, M. Berger and J. L. Kazdan. MR 496885
- Herbert Busemann, Problem IV: Desarguesian spaces, Mathematical developments arising from Hilbert problems (Proc. Sympos. Pure Math., Northern Illinois Univ., De Kalb, Ill., 1974) Amer. Math. Soc., Providence, R. I., 1976, pp. 131–141. Proc. Sympos. Pure Math., Vol. XXVIII. MR 0430935
- Herbert Busemann, Areas in affine spaces. III. The integral geometry of affine area, Rend. Circ. Mat. Palermo (2) 9 (1960), 226–242. MR 136709, DOI https://doi.org/10.1007/BF02854583
- Herbert Busemann, Geometries in which the planes minimize area, Ann. Mat. Pura Appl. (4) 55 (1961), 171–189. MR 143155, DOI https://doi.org/10.1007/BF02412083
- Israel M. Gelfand and Mikhail M. Smirnov, Lagrangians satisfying Crofton formulas, Radon transforms, and nonlocal differentials, Adv. Math. 109 (1994), no. 2, 188–227. MR 1304752, DOI https://doi.org/10.1006/aima.1994.1086
- Felix E. Browder (ed.), Mathematical developments arising from Hilbert problems, American Mathematical Society, Providence, R. I., 1976. MR 0419125
- R. D. Holmes and A. C. Thompson, $n$-dimensional area and content in Minkowski spaces, Pacific J. Math. 85 (1979), no. 1, 77–110. MR 571628
- Aleksei Vasil′evich Pogorelov, Hilbert’s fourth problem, V. H. Winston & Sons, Washington, D.C.; A Halsted Press Book, John Wiley & Sons, New York-Toronto, Ont.-London, 1979. Translated by Richard A. Silverman; Scripta Series in Mathematics. MR 550440
- Rolf Schneider and John André Wieacker, Integral geometry in Minkowski spaces, Adv. Math. 129 (1997), no. 2, 222–260. MR 1462734, DOI https://doi.org/10.1006/aima.1997.1651
- Z. I. Szabó, Hilbert’s fourth problem. I, Adv. in Math. 59 (1986), no. 3, 185–301. MR 835025, DOI https://doi.org/10.1016/0001-8708%2886%2990056-3
- A. C. Thompson, Minkowski geometry, Encyclopedia of Mathematics and its Applications, vol. 63, Cambridge University Press, Cambridge, 1996. MR 1406315
- J.C. Álvarez Paiva, Integral geometry on Finsler manifolds, preprint 1998.
- J.C. Álvarez Paiva, Anti-self-dual symplectic forms and integral geometry, preprint 1998.
- J.C. Álvarez Paiva, I.M. Gelfand, and M. Smirnov, Crofton densities, symplectic geometry, and Hilbert’s fourth problem, in “Arnold-Gelfand Mathematical Seminars, Geometry and Singularity Theory", V.I. Arnold, I.M. Gelfand, and V.S. Retakh (eds.), Birkhäuser, Boston, 1997, pp. 77–92.
- V.I. Arnold and A.B. Givental, Symplectic geometry, in “Dynamical Systems IV", Encyclopaedia of Mathematical Sciences, Vol. 4, V.I. Arnold and S.P. Novikov (Eds.), Springer-Verlag, Berlin, Heidelberg, New York, 1990.
- A. Besse, “Manifolds All of Whose Geodesics are Closed", Springer-Verlag, Berlin, Heidelberg, New York, 1978.
- H. Busemann, Problem IV: Desarguesian spaces, in “Mathematical developments arising from Hilbert problems", Proceedings of Symposia in Pure Math., Vol. XXVII, Part 1, F. Browder (Ed.), AMS, Providence, Rhode Island, 1974.
- H. Busemann, Areas in affine spaces III. The integral geometry of affine area, Rend. Circ. Mat. Palermo (2) 9 (1960), 226–242.
- H. Busemann, Geometries in which the planes minimize area, Ann. Mat. Pura Appl. (4) 55 (1961), 171–190.
- I.M. Gelfand and M. Smirnov, Lagrangians satisfying Crofton formulas, Radon transforms, and nonlocal differentials, Adv. in Math. 109 (1994), 188–227.
- D. Hilbert, Mathematical problems, translation in “Mathematical developments arising from Hilbert problems", Proceedings of Symposia in Pure Math., Vol. XXVII, Part 1, F. Browder (Ed.), AMS, Providence, Rhode Island, 1974.
- R.D. Holmes and A.C. Thompson, N-dimensional area and content in Minkowski spaces, Pacific J. Math. 85 (1979), 77–110.
- A.V. Pogorelov, “Hilbert’s Fourth Problem", Scripta Series in Mathematics, Winston and Sons, 1979.
- R. Schneider and J.A. Wieacker, Integral geometry in Minkowski spaces, Adv. in Math. 129 (1997), 222–260.
- Z.I. Szabo, Hilbert’s fourth problem, I, Adv. in Math. 59 (1986), 185–301.
- A.C. Thompson, “Minkowski Geometry", Encyclopedia of Math. and Its Applications, Vol. 63, Cambridge Univ. Press, Cambridge, 1996.
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Additional Information
J. C. Álvarez Paiva
Affiliation:
Université Catholique de Louvain, Institut de Mathématique Pure et Appl., Chemin du Cyclotron 2, B-1348 Louvain-la-Neuve, Belgium
Email:
alvarez@agel.ucl.ac.be
E. Fernandes
Affiliation:
Université Catholique de Louvain, Institut de Mathématique Pure et Appl., Chemin du Cyclotron 2, B-1348 Louvain-la-Neuve, Belgium
Email:
fernandes@agel.ucl.ac.be
Keywords:
Crofton formulas,
Hilbert’s fourth problem,
Finsler geometry
Received by editor(s):
August 8, 1998
Published electronically:
November 23, 1998
Additional Notes:
Partially supported by a credit aux chercheurs from the FNRS
Communicated by:
Dmitri Burago
Article copyright:
© Copyright 1998
American Mathematical Society
