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The crystal bases are quite useful combinatorial tools to study therepresentations of quantized universal enveloping algebras $U_q(\mathfrak{g})$.The polyhedral realization for $B(\infty)$ is a combinatorial description ofthe 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 $\mathbb{Z}^{\infty}_{\iota}$ isan infinite $\mathbb{Z}$-lattice with a crystal structure associated with$\iota$. It is a natural problem to find an explicit form of the polyhedralrealization ${\rm Im}(\Psi_{\iota})$. 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 thecondition of `adaptedness', we describe ${\rm Im}(\Psi_{\iota})$ by usingseveral combinatorial objects such as extended Young diagrams and Young walls.

Polyhedral realizations for $B(\infty)$ and extended Young diagrams,  Young walls of type ${\rm A}^{(1)}_{n-1}$, ${\rm C}^{(1)}_{n-1}$, ${\rm  A}^{(2)}_{2n-2}$, ${\rm D}^{(2)}_{n}$