We prove the following theorem: If $H$ is a slim hyperbola that
contains a closed set $\mathcal{S}$ of lines in the Euclidean plane,
there exists exactly one hyperbola $H_\min$ of minimal volume that
contains $\mathcal{S}$ and is contained in $H$. The precise concepts
of ``slim'', the ``volume of a hyperbola'' and ``straight lines or
hyperbolas being contained in a hyperbola'' are defined in the text.