  The crystal bases are quite useful combinatorial tools to study the
representations of quantized universal enveloping algebras $U_q(\mathfrak{g})$.
The polyhedral realization for $B(\infty)$ is a combinatorial description of
the crystal base, which is defined as an image of embedding
$\Psi_{\iota}:B(\infty)\hookrightarrow \mathbb{Z}^{\infty}_{\iota}$, where
$\iota$ is an infinite sequence of indices and $\mathbb{Z}^{\infty}_{\iota}$ is
an infinite $\mathbb{Z}$-lattice with a crystal structure associated with
$\iota$. It is a natural problem to find an explicit form of the polyhedral
realization ${\rm Im}(\Psi_{\iota})$. In this article, supposing that
$\mathfrak{g}$ is of affine type ${\rm A}^{(1)}_{n-1}$, ${\rm C}^{(1)}_{n-1}$,
${\rm A}^{(2)}_{2n-2}$ or ${\rm D}^{(2)}_{n}$ and $\iota$ satisfies the
condition of `adaptedness', we describe ${\rm Im}(\Psi_{\iota})$ by using
several combinatorial objects such as extended Young diagrams and Young walls.
