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 compactly supported measure μ , with \hbox{$\mu\left(\R^d\right)>0$} for p ≥ 1 and λ > 0 we consider the functional \begin{equation} E(\gm) = \int_{\R^d} d(x, \Gamma_\gamma)^p {\rm d}\mu(x) + \lambda \,\text{Length}(\gamma) \end{equation} E ( γ ) = ∫ R d d ( x, Γ γ ) p d μ ( x ) + λ Length ( γ ) where γ : I → ℝ d , I is an interval in ℝ, 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 ℋ 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 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.