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arXiv · 2608.10241

Geometry of the subgaussian body of an isotropic convex body

Apostolos Giannopoulos·Minas Pafis·Natalia Tziotziou
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Abstract

For a centered convex body $K\subset\mathbb{R}^n$, let $\Psi_2(K)$ denote the symmetric convex body whose support function is given by the $\psi_2$-norms of linear functionals on $K$. The recent solution of Milman's problem on the existence of subgaussian directions by Letwin and Mikulincer naturally motivates the study of the geometry of this body. We prove that $\Psi_2(K)$ has bounded volume ratio with respect to the $L_2$-centroid body $Z_2(K)$. In the isotropic case, we also obtain sharp estimates for its mean width and the volume radii of its orthogonal projections, and derive consequences for the existence of subgaussian orthonormal bases. In particular, we construct orthonormal bases with quantitatively controlled subgaussian constants for every isotropic convex body.

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Apostolos Giannopoulos, Minas Pafis, Natalia Tziotziou. 2026-08-10. Geometry of the subgaussian body of an isotropic convex body. https://arxiv.org/abs/2608.10241

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