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Self‐contracted curves have finite length
Author(s) -
Stepanov Eugene,
Teplitskaya Yana
Publication year - 2017
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1112/jlms.12068
Subject(s) - mathematics , bounded function , euclidean space , trace (psycholinguistics) , planar , combinatorics , norm (philosophy) , space (punctuation) , metric space , euclidean distance , discrete mathematics , mathematical analysis , geometry , computer science , philosophy , linguistics , computer graphics (images) , political science , law , operating system
A curve θ : I → E in a metric space E equipped with the distance d , where I ⊂ R is a (possibly unbounded) interval, is called self‐contracted, if for any triple of instances of time{ t i } i = 1 3 ⊂ I witht 1 ⩽ t 2 ⩽ t 3one has d ( θ ( t 3 ) , θ ( t 2 ) ) ⩽ d ( θ ( t 3 ) , θ ( t 1 ) ) . We prove that if E is a finite‐dimensional normed space with an arbitrary norm, the trace of θ is bounded, then θ has finite length, that is, is rectifiable, thus answering positively the question raised in Lemenant's paper [‘Rectifiability of non Euclidean planar self‐contracted curves’, Confluentes Math . 8 (2016) 23–38].