Calculus of Variations and Geometric Measure Theory

M. Miranda Jr - E. Paolini - E. Stepanov

On one-dimensional continua uniformly approximating planar sets

created on 15 Feb 2004
modified by paolini on 09 Dec 2017


Published Paper

Inserted: 15 feb 2004
Last Updated: 9 dec 2017

Journal: Calc. Var. Partial Diff. Equations
Volume: 27
Number: 3
Pages: 287-309
Year: 2006


Consider the class of closed connected sets $\Sigma\subset R^n$ satisfying length constraint ${\cal H}(\Sigma)\leq l$ with given $l>0$. The paper is concerned with the properties of minimizers of the uniform distance $F_M$ of $\Sigma$ to a given compact set $M\subset R^n$, \[ F_M(\Sigma):= \max_{y\in M}dist(y,\Sigma), \] where $dist(y, \Sigma)$ stands for the distance between $y$ and $\Sigma$. The paper deals with the planar case $n=2$. In this case it is proven that the minimizers (apart trivial cases) cannot contain closed loops. Further, some mild regularity properties as well as structure of minimizers is studied.

Keywords: urban planning, transport network, multicenter problem, multimedian problem