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GELFAND TRANSFORMS AND CROFTON FORMULAS J.C. ALVAREZ PAIVA AND E. FERNANDES
 

Summary: GELFAND TRANSFORMS AND CROFTON FORMULAS
J.C. ´ALVAREZ PAIVA AND E. FERNANDES
Abstract. 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.
Pensar es olvidar diferencias ... .
Jorge Luis Borges.
Contents
1. Introduction 1
2. Preliminaries 3
3. Gelfand transforms associated to double fibrations 6

  

Source: Alvarez, Juan Carlos - Department of Mathematics, Polytechnic University

 

Collections: Mathematics