1947/10/01 by Herbert Busemann · 6 citations
Physics and Astronomy · Earth and Planetary Sciences · #Advanced Differential Geometry Research #Geophysics and Gravity Measurements
paper · doi:10.2307/2371807
The slow progress in the theory of Finsler spaces as compared to Riemann spaces is partly due to lack of information regarding the corresponding local, that is the Minkowskian, geometry. Those Minkowskian features will contribute most to an understanding of Finsler spaces which are not merely verbal generalizations of known euclidean statements.' The purpose of the present note was originally only to show that the isoperimetric problem (for any dimension) in Minkowski spaces leads to such a feature. It turned out, however, that the plane problem can be solved in a general form-no longer significant for Finsler spaces-and then exhibits a phenomenon which is of interest for the theory of isoperimetric problems in the calculus of variations. The result seems to indicate that the standard methods may have followed too closely the pattern of the fixed endpoint problem. For that reason the plane case is here presented separately. The following are the results: let F(x,,y)be continuous, positive for x, y # 0, and positive homogeneous of order 1. The problem, to find among all simple closed curves x (t), y (t) with a given orientation and a given Minkowski length L = 5 F (x, y) dt one which bounds the greatest (euclidean) area, has a unique solution (up to translations) no matter whether the indicatrix C : F(x, y) =1 is convex or not. For non-convex C t-he solution is the same as for the boundary 0 of the convex closure of C as indicatrix and is homothetic to the polar reciprocal (figuratrix) of 0 with respect to the unit circle rotated through ? 7r/2. For Finsler spaces only the case is of interest where C is convex and has the origin as center. Since Finsler or Minkowski area differs from the euclidean area by a constant factor, the solution of the Minkovskian isoperimetric problem is the same as for the above problem. In intrinsic Minkowskian terms it may be described as the curve of length L for which as-new