Geometric Bounds for the Mean Gauge and the Mean Width in Isotropic Position

Abstract

Let $K\subset {\mathbb{R}}^n$ be an origin-symmetric convex body and assume that its uniform probability measure is isotropic in the probabilistic normalization, namely \[ \int_K x \otimes x \, dμ_K(x) = \mathrm{Id}_n. \] We give deterministic geometric proofs of \[ M(K) \leq C \frac{\log(n)}{\sqrt{n}} \qquad \text{and} \qquad M^*(K) \leq C \sqrt{n} \, \log(n), \] where \[ M(K) = \int_{\mathbb{S}^{n-1}} \|θ\|_K \, dσ(θ), \qquad M^*(K) = \int_{\mathbb{S}^{n-1}} h_K(θ) \, dσ(θ). \] Combining both estimates yields \[ M(K) M^*(K) \leq C \log^2(n). \] The first proof uses a quadratic aggregate of dyadic centroid bodies. The second uses the analogous weighted aggregate of the Laplace bodies $p\{{\Lambda }_K\le p{\}}^{\circ }$, which are equivalent to the centroid bodies by the work of Klartag and E. Milman. In both cases, curvature at each dyadic scale outside a subspace of codimension $O(p)$ leads, via the min--max principle, Legendre duality, and the spherical Laplacian, to the required estimate. The only high-dimensional input is the dimension-free small-ball consequence of the slicing theorem.