  We show an area law with logarithmic correction for the maximally mixed state
$\Omega$ in the (degenerate) ground space of a 1D gapped local Hamiltonian $H$,
which is independent of the underlying ground space degeneracy. Formally, for
$\varepsilon>0$ and a bi-partition $L\cup L^c$ of the 1D lattice, we show that
  $$\mathrm{I}^{\varepsilon}_{\max}(L:L^c)_{\Omega} \leq
O(\log(|L|)+\log(1/\varepsilon)),$$
  where $|L|$ represents the number of qudits in $L$ and
$\mathrm{I}^{\epsilon}_{\max}(L:L^c)_{\Omega}$ represents the $\varepsilon$-
'smoothed maximum mutual information' with respect to the $L:L^c$ partition in
$\Omega$. As a corollary, we get an area law for the mutual information of the
form $\mathrm{I}(L:R)_\Omega \leq O(\log |L|)$. In addition, we show that
$\Omega$ can be approximated up to an $\varepsilon$ in trace norm with a state
of Schmidt rank of at most $\mathrm{poly}(|L|/\varepsilon)$.
