Bourgain's slicing problem and KLS isoperimetry up to polylog (Q2088129)

One of the central questions in high-dimensional convex geometry is Bourgain's slicing problem. It asks whether for any convex body \(K\subset\mathbb{R}^n\) of volume one, there exists a hyperplane \(H\subset\mathbb{R}^n\) such that \(\mathrm{Vol}_{n-1}(K \cap H) > c\), where \(c > 0\) is a constant, and \(\mathrm{Vol}_{n-1}\) stands for \((n - 1)\)-dimensional volume. For \(n \ge 1\) define \[ (L_n)^{-1}:= \inf_{K\subset\mathbb{R}^n}\, \sup_{H\subset\mathbb{R}^n} \mathrm{Vol}_{n - 1}(K \cap H), \] where the infimum runs over all convex bodies of volume one, and the supremum runs over all hyperplanes. A recent breakthrough by \textit{Y. Chen} [Geom. Funct. Anal. 31, No. 1, 34--61 (2021; Zbl 1495.52003)] has led to the bound \[ L_n \le C_1 \exp\left(C_2 \sqrt{\log n}\times \sqrt{\log \log(3n)}\right), \] where \(C_1\), \(C_2 > 0\) are constants. A probability density \(\rho:\mathbb{R}^n \to [0, \infty)\) is log-concave if the set \(\{\rho > 0\} = \{x \in\mathbb{R}^n; \rho(x) > 0\}\) is convex, and \(\log \rho\) is concave in \(\{\rho > 0\}\). A probability measure in \(\mathbb{R}^n\) is log-concave if it is supported in an affine subspace of \(\mathbb{R}^n\) and it has a log-concave density in this subspace. For example, the uniform probability measure on any compact, convex set is log-concave, as well as all Gaussian measures. A probability measure \(\mu\) on \(\mathbb{R}^n\) with finite second moments is isotropic if it is centered and its covariance matrix is the identity matrix. The thin-shell constant \(\sigma_{\mu} > 0\) of an isotropic, log-concave probability measure \(\mu\) in \(\mathbb{R}^n\) is defined via \(n\, \sigma^2_{\mu} = \mathrm{Var}_{\mu}(|x|^2)\). It may be shown that most of the mass of the measure \(\mu\) is located in a spherical shell whose width is at most \(C\, \sigma_{\mu}\), and this estimate for the width is always tight, hence the name thin-shell constant. The parameter \(\sigma_n\) is defined as \(\sigma_n = \sup_{\mu} \sigma_{\mu}\), where the supremum runs over all isotropic, log-concave probability measures \(\mu\) in \(\mathbb{R}^n\). Given a probability measure \(\mu\) in \(\mathbb{R}^n\) with log-concave density \(\rho\), its isoperimetric constant is \[ (\psi_{\mu})^{-1}=\inf_{A\subset\mathbb{R}^n}\left[(\min\{\mu(A), 1 - \mu(A)\})^{-1}\times \int_{\partial A}\rho\right], \] where the infimum runs over all open sets \(A\) with smooth boundary for which \(0 < \mu(A) < 1\). Define \(\psi_n:=\sup_{\mu} \psi_{\mu}\), where the supremum runs over all isotropic, log-concave probability measures \(\mu\) in \(\mathbb{R}^n\). In [loc. cit.], Chen proved the following inequality \[ \psi_n \le C_1 \exp\left(C_2 \sqrt{\log n} \times \sqrt{\log \log(3n)}\right) \] (the same inequality as for \(L_n\)). The main result of this paper is the following statement. Theorem. For any \(n \ge 2\), \(\psi_n \le C(\log n)^5\) for some constant \(C\).