Rectifiability of non Euclidean planar self-contracted curves
Confluentes Mathematici, Volume 8 (2016) no. 2, pp. 23-38.

We prove that any self-contracted curve in 2 endowed with a C 2 and strictly convex norm, has finite length. The proof follows from the study of the curve bisector of two points in 2 for a general norm together with an adaptation of the argument used in [2].

Received:
Revised:
Accepted:
Published online:
DOI: 10.5802/cml.31
Classification: 53A04, 37N40, 49J52, 49J53, 52A10, 65K10
Keywords: Self-contracted curve, uniformly convex norm, rectifiable curve, self-expanded curve, proximal algorithm
Antoine Lemenant 1

1 Université Paris Diderot – Paris 7, CNRS, UMR 7598 Laboratoire Jacques-Louis Lions, Paris, F-75005, France
@article{CML_2016__8_2_23_0,
     author = {Antoine Lemenant},
     title = {Rectifiability of non {Euclidean} planar self-contracted curves},
     journal = {Confluentes Mathematici},
     pages = {23--38},
     publisher = {Institut Camille Jordan},
     volume = {8},
     number = {2},
     year = {2016},
     doi = {10.5802/cml.31},
     language = {en},
     url = {https://cml.centre-mersenne.org/articles/10.5802/cml.31/}
}
TY  - JOUR
AU  - Antoine Lemenant
TI  - Rectifiability of non Euclidean planar self-contracted curves
JO  - Confluentes Mathematici
PY  - 2016
SP  - 23
EP  - 38
VL  - 8
IS  - 2
PB  - Institut Camille Jordan
UR  - https://cml.centre-mersenne.org/articles/10.5802/cml.31/
DO  - 10.5802/cml.31
LA  - en
ID  - CML_2016__8_2_23_0
ER  - 
%0 Journal Article
%A Antoine Lemenant
%T Rectifiability of non Euclidean planar self-contracted curves
%J Confluentes Mathematici
%D 2016
%P 23-38
%V 8
%N 2
%I Institut Camille Jordan
%U https://cml.centre-mersenne.org/articles/10.5802/cml.31/
%R 10.5802/cml.31
%G en
%F CML_2016__8_2_23_0
Antoine Lemenant. Rectifiability of non Euclidean planar self-contracted curves. Confluentes Mathematici, Volume 8 (2016) no. 2, pp. 23-38. doi : 10.5802/cml.31. https://cml.centre-mersenne.org/articles/10.5802/cml.31/

[1] J. Bolte, A. Daniilidis, O. Ley and L. Mazet, Characterizations of Łojasiewicz inequalities: subgradient flows, talweg, convexity, Trans. Amer. Math. Soc. 362 (2010), 3319–3363.

[2] A. Daniilidis, G. David, E. Durand-Cartagena and A. Lemenant Rectifiability of Self-contracted curves in the euclidean space and applications. J. Geom. Anal. 25 (2015), no. 2, 1211–1239.

[3] A. Daniilidis, R. Deville, E. Durand-Cartagena and L. Rifford Self contracted curves in Riemannian manifolds. Preprint. (2015)

[4] A. Daniilidis and Y. Garcia Ramos, Some remarks on the class of continuous (semi-)strictly quasiconvex functions, J. Optim. Theory Appl. 133 (2007), 37–48.

[5] A. Daniilidis, O. Ley and S. Sabourau, Asymptotic behaviour of self-contracted planar curves and gradient orbits of convex functions, J. Math. Pures Appl. 94 (2010), 183–199.

[6] S. Łojasiewicz, Une propriété topologique des sous-ensembles analytiques réels, in: Les Équations aux Dérivées Partielles, pp. 87–89, Éditions du centre National de la Recherche Scientifique, Paris, 1963.

[7] P. Manselli and C. Pucci, Maximum length of steepest descent curves for quasi-convex functions, Geom. Dedicata 38 (1991), 211–227.

[8] J. Palis and W. De Melo, Geometric theory of dynamical systems. An introduction, (Translated from the Portuguese by A. K. Manning), Springer-Verlag, New York-Berlin, 1982.

Cited by Sources: