Sharp Minimax Gram-Determinant Asymptotics for Labelled Unit Decompositions of the Identity and Recent Developments on Makai's Conjecture
Year: 2026
Type: preprint
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.
Source: arXiv (Cornell University)
Author: Zhang, Haozhou
Institution: University of Idaho
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7221096523