Date of Original Version
This is the accepted version of the article which has been published in final form at http://dx.doi.org/10.1051/cocv/2015011
Abstract or Description
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(R d ) > 0 for p ≥ 1 and λ > 0 we consider the functional
E(γ) = Z Rd d(x, Γγ) p dµ(x) + λ Length(γ)
where γ : I → R d , I is an interval in R, Γγ = γ(I), and d(x, Γγ) is the distance of x to Γγ. The problem is closely related to the average-distance problem, where the admissible class are the connected sets of finite Hausdorff measure H1 , 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 p ≥ 2 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.
ESAIM: Control, Optimisation and Calculus of Variations, forthcoming.