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Publication 47

Counterexample to Regularity in Average-Distance Problem

Authors:

CMUDejan Slepčev
Department of Mathematical Sciences
Carnegie Mellon University
Pittsburgh, PA 15213


Abstract:
The average-distance problem is to find the best way to approximate (or represent) a given measure $\mu$ on $\bf{R}^d$ by a one-dimensional object. In the penalized form the problem can be stated as follows: given a finite, compactly supported, positive Borel measure $\mu$, minimize \[ E(\Sigma) = \int_{\bf{R}^d} d(x, \Sigma) d\mu(x) + \lambda \cal{H}^1(\Sigma) \] among connected closed sets, $\Sigma$, where $\lambda >0$, $d(x, \Sigma)$ is the distance from $x$ to the set $\Sigma$, and $\cal{H}^1$ is the one-dimensional Hausdorff measure. Here we provide, for any $d \geq 2$, an example of a measure $\mu$ with smooth density, and convex, compact support, such that the global minimizer of the functional is a rectifiable curve which is not $C^1$. We also provide a similar example for the constrained form of the average-distance problem.
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counterslepcev.pdf

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