  We present two results characterizing minimizers of the Chan-Esedoglu L1TV
functional $F(u) \equiv \int |\nabla 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 $E(\Sigma) = \Per(\Sigma) + \lambda |\Sigma\vartriangle \Omega |$. We show
that there is a minimizer $\Sigma$ such that its boundary $\partial\Sigma$ 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.
