Polyhedral realizations for crystal bases and Young walls of classical affine types

For affine Lie algebra $\mathfrak{g}$ of type $A_{n-1}^{(1)}$, $B_{n-1}^{(1)}$, $C_{n-1}^{(1)}$, $D_{n-1}^{(1)}$, $A_{2n-2}^{(2)}$, $A_{2n-3}^{(2)}$ or $D_n^{(2)}$, let $B(\lambda )$ and $B(\mathrm{\infty })$ be the crystal bases of integrable highest weight representation $V(\lambda )$ and negative part $U_q^{-}(\mathfrak{g})$ of quantum group $U_q(\mathfrak{g})$. We consider the polyhedral realizations of crystal bases, which realize $B(\lambda )$ and $B(\mathrm{\infty })$ as sets of integer points of some polytopes and cones in ${\mathbb{R}}^{\mathrm{\infty }}$. It is a natural problem to find explicit forms of the polytopes and cones. In this paper, we introduce pairs of truncated walls, which are defined as modifications of level $2$-Young walls and describe inequalities defining the polytopes and cones in terms of level $1$-proper Young walls and pairs of truncated walls. As an application, we also give combinatorial descriptions of ${\epsilon }_k^{\ast }$-functions on $B(\mathrm{\infty })$ in terms of Young walls and truncated walls.