Calculus of Variations and Geometric Measure Theory
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X. Y. Lu

Regularity of densities in relaxed and penalized average distance problem

created by lu on 11 Mar 2014
modified on 08 Sep 2015


Accepted Paper

Inserted: 11 mar 2014
Last Updated: 8 sep 2015

Journal: Netw. Heterog. Media
Year: 2015


The average distance problem finds application in data parameterization, which involves ``representing'' the data using lower dimensional objects. From a computational point of view it is often convenient to restrict the unknown to the family of parameterized curves. However this formulation exhibits several undesirable properties. In this paper we propose an alternative variant: the average distance functional is replaced by a transport cost, and the unknown is composed both by a curve and by a ``projected measure'', on which an $L^q$ penalization term is added. Moreover we will add a term penalizing non injectivity. We will use techniques from optimal transport theory and calculus of variations. The main aim is to prove essential boundedness, and a variant of Lipschitz continuity for Radon-Nikodym derivative of projected measures for minimizers.


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