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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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.