On the asymptotics of the principal moments of inertia of a convex body in the isotropic state (Q2246236)

This paper contains a short discussion of Bourgain's conjecture on a universal upper bound for the isotropic constant \(L_K\) of an isotropic convex body \(K\). The author calculates the isotropic constant for the unit cube, \(I^n\), and furthermore its asymptotics for the ball of unit volume, \(b^n\). These examples also corroborate the famous slicing problem or hyperplane conjecture since they are equivalent to the question of a universal upper bound for the isotropic constant. In Section 2, the author shows that \(L_{b^n}^2 \simeq \frac{1}{2\pi e}\) as \(n\to \infty\). In Section 3, the author obtains that \(L_{I^n}^2 = \frac{1}{12}\) for any dimension \(n\). In Section 4, these results are interpreted under the aspect of concentration of measure.