On the isotropy constant of projections of polytopes (Q849002)

A convex body \(K\subset\mathbb{R}^n\) is isotropic if it has volume \(\text{Vol}_n(K)= 1\), the barycenter of \(K\) is at the origin and its inertia matrix is a multiple of the identity. Equivalently, there exists a constant \(L_K> 0\) called isotropy constant of \(K\) such that \(L^2_K= \int_K\langle x,0\rangle^2\,dx\), for all \(\theta\in S^{n-1}\). The boundedness of the isotropy constant is a major conjecture in asymptotic geometric analysis. The answer is known to be positive for many families of convex bodies. In this paper, the authors focus their attention on the isotropy constant of polytopes or, equivalently, of projections of the unit ball of \(\ell^n_1\) space (in the symmetric case) and of the regular \(n\)-dimensional simplex \(S_n\) (in the non-symmetric case). The main result of the paper states that for any \(d\)-dimensional polytope \(K\) with \(n\) vertices its isotropy constant \(L_K\) satisfies \(L_K\leq C\sqrt{{n\over d}}\), where \(C> 0\) is a numerical constant.