Average-distance problem for parameterized curves
ESAIM: Control, Optimisation and Calculus of Variations, Tome 22 (2016) no. 2, pp. 404-416.

We consider approximating a measure by a parameterized curve subject to length penalization. That is for a given finite compactly supported measure μ, with μ d >0 for p1 and λ>0 we consider the functional

E()= d d(x,Γ γ ) p dμ(x)+λLength(γ)

where γ:I d , I is an interval in , Γ γ =γ(I), and d(x,Γ γ ) is the distance of x to Γ γ . The problem is closely related to the average-distance problem, where the admissible class are the connected sets of finite Hausdorff measure 1 , and to (regularized) principal curves studied in statistics. We obtain regularity of minimizers in the form of estimates on the total curvature of the minimizers. We prove that for measures μ supported in two dimensions the minimizing curve is injective if p2 or if μ has bounded density. This establishes that the minimization over parameterized curves is equivalent to minimizing over embedded curves and thus confirms that the problem has a geometric interpretation.

Reçu le :
DOI : 10.1051/cocv/2015011
Classification : 49Q20, 49K10, 49Q10, 35B65
Mots clés : Average-distance problem, principal curves, nonlocal variational problems
Lu, Xin Yang 1 ; Slepčev, Dejan 1

1 Department of Mathematical Sciences, Carnegie Mellon University, Pittsburgh, PA, 15213, USA
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Lu, Xin Yang; Slepčev, Dejan. Average-distance problem for parameterized curves. ESAIM: Control, Optimisation and Calculus of Variations, Tome 22 (2016) no. 2, pp. 404-416. doi : 10.1051/cocv/2015011. http://archive.numdam.org/articles/10.1051/cocv/2015011/

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