Rectifiability Property for Plane Paths and Descent Curves
- PDF / 292,819 Bytes
- 9 Pages / 439.37 x 666.142 pts Page_size
- 0 Downloads / 163 Views
Rectifiability Property for Plane Paths and Descent Curves Nico Lombardi1 · Marco Longinetti2
· Paolo Manselli2 · Adriana Venturi3
Received: 10 April 2020 / Accepted: 13 August 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract A family of plane, oriented and continuous paths depending on a fixed real positive number is considered. For any point on the path, the previous points lie out of any disk with same radius, having interior normal in a suitable tangent cone to the path. These paths are locally descent curves of a nested family sets of same positive reach. Avoiding any smoothness requirements, we get angle estimate and not intersection property. Afterwards, we are able to estimate the length and detour of this curve. Keywords Steepest descent curves · Sets with positive reach · Length of curves · Detour Mathematics Subject Classification Primary: 28A75 · 52A30 · Secondary: 34A26 · 34A60
1 Introduction Let g be a plane and continuous mapping, oriented according to the increasing variable. Let us assume that for any point on the support of the map, there is a constraint for g; then what extra properties does the mapping g satisfy? Different constraints for the previous points have been considered by several authors, and for each constraint different classes of family of curves have been studied. Most of these curves are related to descent curves, as in [1] and [2]. Let us list some interesting examples:
Communicated by Aris Daniilidis.
B
Marco Longinetti [email protected]
1
Institute of Discrete Mathematics and Geometry, Vienna University of Technology, Wiedner Hauptstraße 8–10/104, 1040 Wien, Austria
2
DIMAI (Department of Mathematics and Computer Science, Ulisse Dini), Università di Firenze, V.le Morgagni 67/a, 50134 Firenze, Italy
3
Università di Firenze, P.le delle Cascine 15, 50144 Firenze, Italy
123
Journal of Optimization Theory and Applications
Example 1.1 Let the constraint be as follows: for any three points on the support of the map, the distance between the first two consecutive points is less than the distance between the first and the third one. This family of mappings has been studied in [3] as steepest descent curves of quasi-convex functions, and they have been studied in several dimensions and called self-expanding curves in [4]. In [4] and [5], it was proved that g is rectifiable: as a consequence, if x(s) is a point on the support of the curve, parametrized in arc length, then it turns out that the previous points to x(s) lie out of the half-space orthogonal to x (s) at x. It turns out that the continuous mapping g is an injective rectifiable curve and a bound for its length is given. Moreover, there exists a quasi-convex function f and a nested family of convex sets, such that the boundary of each convex set of the family is a level set for f and the curve g is a steepest descent curve for f . Let us remark that continuous mapping g, with similar constraint, was called self-approaching curves in [6], self-contracting curves in [5] a
Data Loading...
