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Journal of Applied Mathematics
Volume 2016 (2016), Article ID 4873276, 17 pages
http://dx.doi.org/10.1155/2016/4873276
Research Article

Bounding Regions to Plane Steepest Descent Curves of Quasiconvex Families

1Dipartimento di Matematica e Informatica Ulisse Dini, Università degli Studi di Firenze, Viale Morgagni 67/a, 50134 Firenze, Italy
2Dipartimento GESAAF, Università degli Studi di Firenze, Piazzale delle Cascine 15, 50144 Firenze, Italy

Received 26 February 2016; Accepted 17 April 2016

Academic Editor: Wenyu Sun

Copyright © 2016 Marco Longinetti et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Two-dimensional steepest descent curves (SDC) for a quasiconvex family are considered; the problem of their extensions (with constraints) outside of a convex body is studied. It is shown that possible extensions are constrained to lie inside of suitable bounding regions depending on . These regions are bounded by arcs of involutes of and satisfy many inclusions properties. The involutes of the boundary of an arbitrary plane convex body are defined and written by their support function. Extensions SDC of minimal length are constructed. Self-contracting sets (with opposite orientation) are considered: necessary and/or sufficient conditions for them to be subsets of SDC are proved.