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PreprintarXiv licencearXiv

Sharp Minimax Gram-Determinant Asymptotics for Labelled Unit Decompositions of the Identity and Recent Developments on Makai's Conjecture

  • Haozhou Zhang
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Abstract

We show that González Merino and Schymura's Conjecture 4.8, a sufficient condition for Makai's reverse isodiametric conjecture, fails in all sufficiently large dimensions. Its three-dimensional case follows from Aliev's proof. Pełczyński and Szarek's construction for enclosing parallelepipeds implies this failure. Using random frames, we also determine the sharp exponential determinant guarantee for every fixed ratio greater than one between the number of vectors and the dimension.

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Cite

BibTeXRIS
BibTeX
@misc{zhang2026sharp,
  title = {{Sharp Minimax Gram-Determinant Asymptotics for Labelled Unit Decompositions of the Identity and Recent Developments on Makai's Conjecture}},
  author = {Haozhou Zhang},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.10979},
  eprint = {2610.10979},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.10979},
}
RIS
TY  - UNPB
TI  - Sharp Minimax Gram-Determinant Asymptotics for Labelled Unit Decompositions of the Identity and Recent Developments on Makai's Conjecture
AU  - Haozhou Zhang
PY  - 2026
DA  - 2026/10/09
PB  - arXiv
DO  - 10.48550/arxiv.2610.10979
UR  - https://arxiv.org/abs/2610.10979
AB  - We show that González Merino and Schymura's Conjecture 4.8, a sufficient condition for Makai's reverse isodiametric conjecture, fails in all sufficiently large dimensions. Its three-dimensional case follows from Aliev's proof. Pełczyński and Szarek's construction for enclosing parallelepipeds implies this failure. Using random frames, we also determine the sharp exponential determinant guarantee for every fixed ratio greater than one between the number of vectors and the dimension.
ER  -