An Introduction to Finsler Geometry
by Juan Carlos Álvarez Paiva and Carlos E. Durán
This short book is based on the notes for seven lectures given
by the authors at the "Escuela Venezolana de Matemáticas"
in the summer of 1998. Special emphasis is placed on geometric
concepts, the construction of interesting examples, and the
geometric interpretation of the differential invariants of
Finsler surfaces. This book is not representative of the great
body of work in Finsler geometry in the past 50 years and readers
looking for tensor calculations will be much dissapointed. On the
other hand, convex geometers, globalRiemannian geometers, and
lovers of synthetic, geometric constructions are likely to find it
interesting.
Crofton formulas and Gelfand transforms
by J.C. Álvarez Paiva and E. Fernandes.
The term integral geometry has come to describe two different
fields of research: one, geometrical, based on the works of Blaschke,
Chern, and Santaló, and another, analytical, based on the works of
Radon, John, Helgason, and Gelfand. In this paper the authors bridge the gap
by showing that classical integralgeometric formulas such as those of
Crofton, Cauchy, and Chern can be easily and systematically obtained
through the study of Radontype transforms on double fibrations. The
methods are also used to extend these formulas to nonhomogeneous settings
where grouptheoretic techniques are no longer useful. To illustrate this
point, all Finsler metrics on projective space such that
hyperplanes are areaminimizing are constructed and the theory of Crofton
densities developed by Busemann, Pogorelov, Gelfand, and Smirnov is substantially
simplified.

Some Problems in Finsler Geometry
New Version
by Juan Carlos Álvarez Paiva
From the introduction:
... the present paper includes thirty one simplystated open
problems, as well as a survey of the more elementary and geometric
chapters of Finsler geometry. It presents a detailed discussion of
the notion of volume and area on Finsler manifolds with a strong bias
towards the socalled HolmesThompson definition which, because of its
symplectic nature, is easier to work with than the Hausdorff measure
the other highlights of the paper are its presentation of Hilbert's
fourth problem and its elementary approach to the differential
invariants of Finsler surfaces. [...]
In view of the oftenmade criticisms of Finsler geometry —
very few concrete and interesting examples, very few nonRiemannian
theorems of real geometric content, and too many subindices —
I have tried to include as many concrete examples, simplystated
results, and geometric constructions as possible. In this way,
many of the jewels, so to speak, of Finsler geometry find their
way in to the following pages.
This new version contains a more careful treatment of
of areas in Finsler and Minkowski spaces, the geometry of unit spheres
in normed spaces, and the symplectic equivalence of Finsler manifolds.
I have removed some problems and I have added others which seem more
interesting.
