A series of essentially equivalent statements on zonoids which hold for $d=2$ but fail for $d\geq 3$:

any convex symmetric polytope in $\mathbb{R}^d$ is a Minkowski sum of segments;

any convex symmetric body in $\mathbb{R}^d$ is a section of a unit ball in $L^1$-type space (for polytope finite-dimensional hyperoctahedron is enough);

any convex symmetric body in $\mathbb{R}^d$ is a projection of a unit ball in $L^{\infty}$-type space (for polytope finite-dimensional cube is enough);

any Banach norm in $\mathbb{R}^d$ may be expressed as $\|x\|=\int |(x,y)| d\mu(y)$ for some Borel measure $\mu$ on $\mathbb{R}^d$, where $(x,y)$ is a scalar product;

for any norm $\|\cdot\|$ on $\mathbb{R}^d$ and any vectors $v_1,u_1,\dots,v_n,u_n$ we have $\sum \|u_i\|\geq \sum \|v_i\|$ provided that $\sum |(u_i,y)|\geq \sum |(v_i,y)|$ for any vector $y$.