Extremal Distances between Sections of Convex Bodies

Let K, D be centrally symmetric convex bodies in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb{R}^n .$$\end{document} Let k < n and let dk(K, D) be the smallest Banach–Mazur distance between k-dimensional sections of K and D. Define \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \Delta (k,n) = \sup d_k (K,D), $$\end{document} where the supremum is taken over all n-dimensional convex symmetric bodies K, D. We prove that, for any k < n, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \Delta (k,n) \sim _{{\text{log}}\,n} \left\{ {\begin{array}{*{20}l} {\sqrt k } & {{\text{if}}\;k \leq n^{2/3} ,} \\ {\frac{{k^2 }} {n}} & {{\text{if}}\;k > n^{2/3} ,} \\ \end{array} } \right. $$\end{document} where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A \sim _{\log n} B$$\end{document} means that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$1/(C\log ^a n) \cdot A \leq B \leq (C\log ^a n) \cdot A$$\end{document} for some absolute constants C, a  > 0.