Gelfand transforms and Crofton formulas

• J. C. Alvarez Paiva ^[1] ; E. Fernandes ^[2]
  1. [1] Université des Sciences et Technologies de Lille
  2. [2] MULTITEL Research Centre
• Localización: Selecta Mathematica, New Series, ISSN 1022-1824, Vol. 13, Nº. 3, 2007, págs. 369-390
• Idioma: inglés
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• Resumen

  • The term integral geometry has come to describe two different fields of research: one, geometrical, based on the works of Blaschke, Chern, and Santal´o, and another, analytical, based on the works of Radon, John, Helgason, and Gelfand. In this paper we bridge the gap by showing that classical integral-geometric formulas such as those of Crofton, Cauchy, and Chern can be easily and systematically obtained through the study of Radon-type transforms on double fibrations. The methods also allow us to extend these formulas to non-homogeneous settings where group-theoretic techniques are no longer useful. To illustrate this point, we construct all Finsler metrics on projective space such that hyperplanes are area-minimizing and extend the theory of Crofton densities developed by Busemann, Pogorelov, Gelfand, and Smirnov.