Some properties of minimizers for the Chan-Esedoglu L1TV functional

We present two results characterizing minimizers of the Chan-Esedoglu L1TV functional $F(u) equiv int | abla u | dx + lambda int |u - f| dx $; $u,f:Bbb{R}^n to Bbb{R}$. If we restrict to $u = chi_{Sigma}$ and $f = chi_{Omega}$, $Sigma, Omega in Bbb{R}^n$, the $L^1$TV functional reduces to $E(Sigma) = Per(Sigma) + lambda |Sigmavartriangle Omega |$. We show that there is a minimizer $Sigma$ such that its boundary $partialSigma$ lies between the union of all balls of radius $frac{n}{lambda}$ contained in $Omega$ and the corresponding union of $frac{n}{lambda}$-balls in $Omega^c$. We also show that if a ball of radius $frac{n}{lambda} + epsilon$ is almost contained in $Omega$, a slightly smaller concentric ball can be added to $Sigma$ to get another minimizer. Finally, we comment on recent results Allard has obtained on $L^1$TV minimizers and how these relate to our results.