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 $p \geq 1$ and $λ>0$ we consider the functional \[ E(γ) = \int_{\mathbb{R}^d} d(x, Γ_γ)^p dμ(x) + λ\,\textrm{Length}(γ) \] where $γ:I \to \mathbb{R}^d$, $I$ is an interval in $\mathbb{R}$, $Γ_γ= γ(I)$, 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 $\mathcal H^1$, 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 $p \geq 2$ 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.