Zonoids whose polars are zonoids: the Banach--Mazur distance need not tend to one

Abstract

For every $n\ge 2$, we consider a Gaussian zonoid of revolution $Z_n$ arising from works of Vitale and Mathis. We prove that $Z_n^{\ast }$ is also a zonoid and compute the Banach--Mazur distance from $Z_n$ to the Euclidean ball. This distance is independent of the dimension and is approximately $1.10$. Consequently, the supremal Banach--Mazur distance among zonoids whose polars are zonoids does not converge to $1$. The construction also produces a separable real Banach space $X$, not isometric to a Hilbert space, such that both $X$ and $X^{\ast }$ embed linearly isometrically into $L_1$.