What is wrong with the Hausdorff measure in Finsler spaces

By J. '. Paiva, G. Berck. 2004

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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 Ivanov [On two-dimensional minimal fillings, St. Petersburg Math. J. 13 (2002) 17-25] 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 Finster spaces and give an easy proof that their totally geodesic hypersurfaces are minimal. (C) 2005 Elsevier Inc. All rights reserved.

Subfields: Applied Mathematics, Astronomy and Astrophysics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/e9b18e5e153e675ab94b08ed4d851eb02b277cfc

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In Applied Mathematics

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Citations in the corpus

Minkowski Areas and Valuations (2010, background): “Central questions in the geometry of finite dimensional Banach spaces are those of finding the right notion of Minkowski area and, more generally, of invariant area (see Section 2 for the definition), and of finding distinctive properties of different such areas (see [9] and [55] for more informatio”
Boundary rigidity and filling volume minimality of metrics close to a flat one (2010, background): “Introduce a family of linear maps Lt : X → H, t ∈ [0, 1] by Lt = L+ t ·V .”