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

Dmitry Ryabogin; Artem Zvavitch

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arXiv · 2609.10852

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

Dmitry Ryabogin·Artem Zvavitch

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Abstract

For every $n\geq2$, we consider a Gaussian zonoid of revolution $Z_n$ arising from works of Vitale and Mathis. We prove that $Z_n^*$ 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 $X^*$ embed linearly isometrically into $L_1$.

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BibTeX RIS

Dmitry Ryabogin, Artem Zvavitch. 2026-09-09. Zonoids whose polars are zonoids: the Banach--Mazur distance need not tend to one. https://arxiv.org/abs/2609.10852

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