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A technique for finding minimal paths in subspaces of a metric space

Published online by Cambridge University Press:  09 April 2009

Harold Willis Milnes  and
S. K. Hildebrand
Harold Willis Milnes
Affiliation:
Texas Tech UniversityLubbock, Texas 79409, U.S.A.
S. K. Hildebrand
Affiliation:
H-M Consultants Lubbock, Texas 79409, U.S.A.
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In this paper the problem of constructing an arc of minimum length joining two fixed points: P1, P2, in an arbitrary subset: S, of a metric space is considered. The approach taken is a departure from the classical methods of the calculus of variations in that it is topological character, making use of the properties of sets rather than differential calculus.

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 19 , Issue 3 , May 1975 , pp. 312 - 320
Copyright
Copyright © Australian Mathematical Society 1975

References

Milnes, H. W. and Hildebrand, S. H. (1970), ‘Non-isolated minimising arcs’, SIAM J. Appl. Math. 18, 139–149.CrossRefGoogle Scholar
Milnes, H. W. and Hildebrand, S. K. (1972), ‘Mappings that remove singularities’, Texas J. Science 24, 183–190.Google Scholar
Milnes, H. W. and Hildebrand, S. K. (1972), ‘Maxima and minima of functions and functinals’, Studia Sci. Math. Hungar. 7, 249–256.Google Scholar
Milnes, H. W. and Hildebrand, S. K. (submitted), ‘Endpoint conditions for the problem extremizing a non-parametric integral in the plane’.Google Scholar