Skip to content
DraconiticResearch
Search Sources API
203,234 studies31 sourcesUpdated 9 min ago
PreprintCC BY 4.0arXiv

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

  • Dmitry Ryabogin
  • Artem Zvavitch
Open at sourcePDFDOICite

Abstract

For every n≥2, we consider a Gaussian zonoid of revolution Zn arising from works of Vitale and Mathis. We prove that Zn* is also a zonoid and compute the Banach--Mazur distance from Zn 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 L1.

Related

  1. Projection Constants and Banach-Mazur Distance in Codimension Two
    Beata Deregowska, Barbara Lewandowska
    arXivPreprintCC BY 4.0Full textAnalysis & PDEsarXiv
  2. A 5/8 Lower Bound on the Banach--Mazur Distance to the Cross-Polytope
    Omer Friedland
    arXivPreprintarXiv licenceFull textAnalysis & PDEsProbabilityarXiv
  3. The sharp exponent for the Banach--Mazur distance to the cube
    Omer Friedland
    arXivPreprintarXiv licenceFull textAnalysis & PDEsarXiv
  4. Gaussian polytopes with large Banach-Mazur distance to the cross-polytope
    Antonios Hmadi
    arXivPreprintCC BY-NC-SA 4.0Full textAnalysis & PDEsGeometry & topologyarXiv
  5. Tight Stability Estimates Near the Simplex and Applications for the Banach-Mazur Distance and Rogers-Shephard-Type Inequalities
    René Brandenberg, Bernardo González Merino, Florian Grundbacher
    arXivPreprintCC BY 4.0Full textGeometry & topologyarXiv

Cite

BibTeXRIS
BibTeX
@misc{ryabogin2026zonoids,
  title = {{Zonoids whose polars are zonoids: the Banach--Mazur distance need not tend to one}},
  author = {Dmitry Ryabogin and Artem Zvavitch},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.10852},
  eprint = {2609.10852},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.10852},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Zonoids whose polars are zonoids: the Banach--Mazur distance need not tend to one
AU  - Dmitry Ryabogin
AU  - Artem Zvavitch
PY  - 2026
DA  - 2026/09/09
PB  - arXiv
DO  - 10.48550/arxiv.2609.10852
UR  - https://arxiv.org/abs/2609.10852
AB  - For every n≥2, 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.
ER  -