Fixed-Parameter Tractability of Zonotope Problems

Given a rational generator matrix $G\in\mathbb Q^{d\times n}$ with columns $g_1,\ldots,g_n$, define

$$Z(G)=\sum_{i=1}^n\operatorname{conv}\{0,g_i\} =\left\{\sum_{i=1}^n\lambda_i g_i:0\leq\lambda_i\leq1\right\}.$$

An algorithm is fixed-parameter tractable in $d$ if it runs in $f(d)n^{O(1)}$ time for some computable function $f$.

1. Is $L_p$-maximization over $d$-zonotopes fixed-parameter tractable with respect to $d$ for $p>1$?

2. Is $d$-zonotope containment fixed-parameter tractable with respect to $d$? That is, given generator matrices $G_1,G_2$, decide whether $Z(G_1)\subseteq Z(G_2)$ in $f(d)n^{O(1)}$ time.