Zonoid

Remarks

Equivalently, the support function of $Z$ has the form $h_Z(x) = x \cdot a + \int_{S^{n-1}} \abs{x \cdot v} \, d\mu(v)$. Here $a \in \R^n$, and $\mu$ is a finite even Borel measure on the unit sphere $S^{n-1}$. Example: the closed unit ball $B$ of $\R^n$ is a zonoid. Its support function is $h_B(x) = \abs{x}$, and by rotation invariance $\int_{S^{n-1}} \abs{x \cdot v} \, d\sigma(v) = c_n \abs{x}$ for the normalized surface measure $\sigma$ and a constant $c_n > 0$, so $\mu = \sigma / c_n$ works with $a = 0$. Non-example: a triangle in $\R^2$ is not a zonoid. Each zonotope is symmetric about a point, this property passes to Hausdorff limits, and a triangle has no centre of symmetry. In $\R^2$ every convex body with a centre of symmetry is a zonoid; in $\R^3$ the regular octahedron is a centrally symmetric convex body that is not a zonoid. Some authors also call lower-dimensional Hausdorff limits of zonotopes, such as single segments, zonoids; here a zonoid is required to have nonempty interior, like every convex body.