For any isotropic convex body in R^n, the subgaussian body has bounded volume ratio against the centroid body, sharp mean width O(√log n), and an orthonormal basis with subgaussian constants O(√log n).
Optimal $MM^*$ bounds for convex bodies
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abstract
Let $K\subset\R^n$ be a convex body in isotropic position. We prove the optimal mean-width estimate \[ M^*(K)\leq C\sqrt{n\log n}. \] The main new ingredient is a geometric inequality relating the Gaussian mean of the support function to its mean under the uniform measure on $K$, obtained through a heat-flow argument. Combined with the Gaussian-log-concave comparison of Eldan and Lehec and the newly available dimension-free bound on the third-moment parameter $\kappa_n$, this yields the result. The boundedness of $\kappa_n$ also makes the mean-norm estimate of Bizeul and Klartag sharp. Combining both estimates yields \[ M(K)M^*(K)\leq C\log n, \] extending Pisier's $MM^*$ estimate to non-symmetric convex bodies and to the isotropic position.
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For any measurable fundamental domain K of a lattice Lambda, the flat torus spectral gap obeys lambda_SG(T_Lambda) >= pi^2/(3||Cov_K||_op), with sharp constants and an equivalence under a sectional tiling condition.
New moment comparisons and dyadic chaining yield an n^{1/4} polylog coefficient in Latała's weak-strong inequality for isotropic log-concave vectors, plus sharp L2-Sudakov and polar centroid body entropy estimates.
For origin-symmetric convex bodies in isotropic position, deterministic geometric arguments yield M(K) ≤ C log(n)/√n and M*(K) ≤ C√n log(n), hence MM* ≤ C log² n.
citing papers explorer
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Geometry of the subgaussian body of an isotropic convex body
For any isotropic convex body in R^n, the subgaussian body has bounded volume ratio against the centroid body, sharp mean width O(√log n), and an orthonormal basis with subgaussian constants O(√log n).
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Spectral and Isoperimetric Bounds on Flat Tori
For any measurable fundamental domain K of a lattice Lambda, the flat torus spectral gap obeys lambda_SG(T_Lambda) >= pi^2/(3||Cov_K||_op), with sharp constants and an equivalence under a sectional tiling condition.
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Moment comparisons, Sudakov inequalities and entropy of centroid bodies
New moment comparisons and dyadic chaining yield an n^{1/4} polylog coefficient in Latała's weak-strong inequality for isotropic log-concave vectors, plus sharp L2-Sudakov and polar centroid body entropy estimates.
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Geometric Bounds for the Mean Gauge and the Mean Width in Isotropic Position
For origin-symmetric convex bodies in isotropic position, deterministic geometric arguments yield M(K) ≤ C log(n)/√n and M*(K) ≤ C√n log(n), hence MM* ≤ C log² n.
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