Neighborhoods on the Grassmannian of marginals with bounded isotropic constant
Let \(K\) be a symmetric convex body in \({\mathbb R}^n\) of volume 1. The hyperplane conjecture (see \textit{J. Bourgain} [Lect. Notes Math. 1469, 127--137 (1991; Zbl 0773.46013)]) claims that there exist a universal constant \(c>0\) and a unit vector \(\theta\) such that \(|K\cap \theta^{\bot}|\geq c\), where \(|\cdot|\) stands for the \(n\)-dimensional volume. A Borel probability measure \(\mu\) on \({\mathbb R}^n\) is called log-concave if for any compact sets \(A\), \(B\) in \({\mathbb R}^n\) we have \(\mu((1-\lambda) A + \lambda B) \geq \mu(A)^{1-\lambda} \mu(B)^{\lambda}\) for all \(\lambda\in (0, 1)\). The measure \(\mu\) is called centered if \(\int\langle x, y \rangle\, d\mu(x)=0\) for all \(y\in {\mathbb R}^n\). The covariance matrix for a centered measure \(\mu\) is defined as: \(\text{{Cov}}(\mu)_{i j}=\int_{{\mathbb R}^n} x_i x_j\, d\mu(x)\), \(i, j =1,\dots, n\). A log-concave probability measure \(\mu\) is called isotropic if it is centered and its covariance matrix is the identity. The isotropic constant of a centered log-concave probability measure is defined as: \(L_{\mu} = \|\mu\|_{\infty}^{1/n}\left[\det \text{{Cov}}(\mu)\right]^{{1\over{2n}}}\), where \(\|\mu\|_{\infty} = \|f_{\mu}\|_{\infty}\), where \(f_{\mu}\) is the density of \(\mu\). NEWLINENEWLINEThe hyperplane conjecture can be formulated equivalently as follows: there exists a constant \(C>0\) such that for all \(n\) and any \(\mu\) log-concave isotropic probability measure in \({\mathbb R}^n\), \(L_{\mu}< C\). For a more detailed exposition on recent developments see [\textit{S. Brazitikos} et al., Geometry of isotropic convex bodies. Providence, RI: American Mathematical Society (AMS) (2014; Zbl 1304.52001)]. The first non-trivial bound on this question was given in [Bourgain, loc. cit.] proving that \(L_{\mu}\leq c_1 n^{1/4} \log n\) for all isotropic measures on \({\mathbb R}^n\). In [\textit{B. Klartag}, Geom. Funct. Anal. 16, No. 6, 1274--1290 (2006; Zbl 1113.52014)], the logarithmic term has been removed. For an alternative approach see \textit{B. Klartag} and \textit{E. Milman} [J. Funct. Anal. 262, No. 1, 10--34 (2012; Zbl 1236.52003)]. NEWLINENEWLINEThe paper is organized as follows. In Section 2 the authors give the necessary background for the Grassmann manifold and some facts from convexity. In Section 3 the main result of the paper is proved. In Section 4 the authors conclude with the optimality on the theorem and some remarks.
scientific article; zbMATH DE number 6349103
scientific article