Average-distance problem for parameterized curves

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.