AVERAGE-DISTANCE PROBLEM FOR PARAMETERIZED CURVES

ABSTRACT. We consider approximating a measure by a parameterized curve subject to length penalization. That is for a given finite positive compactly supported measure µ, for p ≥ 1 and λ> 0 we consider the functional E(γ) = Rd d(x,Γγ) pdµ(x) + λLength(γ) where γ: I → Rd, 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