Juan Carlos Alvarez Paiva

arXiv2004

What is wrong with the Hausdorff measure in Finsler spaces

We construct a class of Finsler metrics in three-dimensional space such that all their geodesics are lines, but not all planes are extremal for their Hausdorff area functionals. This shows that if the Hausdorff measure is used as notion of volume on Finsler spaces, then totally geodesic submanifolds are not necessarily minimal, filling results such as those of S. Ivanov do not hold, and integral-geometric formulas do not exist. On the other hand, using the Holmes-Thompson definition of volume, we prove a general Crofton formula for Finsler spaces and give an easy proof that their totally geodesic hypersurfaces are minimal.

math.DG↗

arXiv2004

Dual spheres have the same girth

Symplectic and Finsler geometry are used to settle a conjecture of Schäffer stating that the girth of a normed space--the infimum of the lengths of all closed, rectifiable, centrally symmetric curves on its unit sphere--equals the girth of its dual.

math.MG↗

arXiv2004

Dual mixed volumes and isosystolic inequalities

The theory of dual mixed volumes is extended to star bodies in cotangent bundles and is used to prove several isosystolic inequalities for Hamiltonian systems and Finsler metrics.

math.SG↗