  In this paper we consider a minimization problem which arises from thermal
insulation. A compact connected set $K$, which represents a conductor of
constant temperature, say $1$, is thermally insulated by surrounding it with a
layer of thermal insulator, the open set $\Omega\setminus K$ with
$K\subset\bar\Omega$. The heat dispersion is then obtained as \[ \inf\left\{
\int_{\Omega}|\nabla \varphi|^{2}dx
+\beta\int_{\partial^{*}\Omega}\varphi^{2}d\mathcal H^{n-1} ,\;\varphi\in
H^{1}(\mathbb R^{n}), \, \varphi\ge 1\text{ in } K\right\}, \] for some
positive constant $\beta$. We mostly restrict our analysis to the case of an
insulating layer of constant thickness. We let the set $K$ vary, under
prescribed geometrical constraints, and we look for the best (or worst)
geometry in terms of heat dispersion. We show that under perimeter constraint
the disk in two dimensions is the worst one. The same is true for the ball in
higher dimension but under different constraints. We finally discuss few open
problems.
