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

For affine Lie algebra $\mathfrak{g}$ of type $A^{(1)}_{n-1}$, $B^{(1)}_{n-1}$, $C^{(1)}_{n-1}$, $D^{(1)}_{n-1}$, $A^{(2)}_{2n-2}$, $A^{(2)}_{2n-3}$ or $D^{(2)}_{n}$, let $B(\lambda)$ and $B(\infty)$ 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 $B(\infty)$ as sets of integer points of some polytopes and cones in $\mathbb{R}^{\infty}$. 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 $\varepsilon_k^*$-functions on $B(\infty)$ in terms of Young walls and truncated walls.