Summary
It is shown that for 1≤a≤n an open convex set in n-dimensional affine space can be provided with a complete a-dimensional area for which the a-flats minimize area, and that any such area can be extended to an area with the same propertiés in an open convex set in (n+1)—space.
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References
H. Busemann,The Geometry of Geodesics, New York 1955.
H. Busemann,Convex Surfaces, New York 1958.
D. Hilbert,Mathematische Probleme, « Nachr. Akad. Wiss. Göttingen, Math. Phys. », 1900, pp. 253–297.
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To Enrico Bompiani on his scientific Jubilee.
This work was supported by a grant from the National Science Foundation.
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Busemann, H. Geometries in which the planes minimize area. Annali di Matematica 55, 171–189 (1961). https://doi.org/10.1007/BF02412083
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DOI: https://doi.org/10.1007/BF02412083