High dimensional random sections of isotropic convex bodies
The authors study two properties of random high dimensional convex bodies. In the first part they consider a symmetric convex body \(K\) and they estimate the central section function \(|K\cap F^{\bot}|_{n-k}^{1/k}\), for random \(F\subset G_{n,k}\), \(1\leq k\leq n,\) where \(G_{n,k}\) is the Grassmann space. In [Lect. Notes Math. 1376, 64--104 (1989; Zbl 0679.46012), p. 88], \textit{V. D. Milman} and \textit{A. Pajor} show that \(|K\cap F^{\bot}|_{n-k}^{1/k}\sim L_{B_{k+1}}(K,F)/L_K\), where \(L_K\) is the isotropy constant of \(K\), \(B_{k+1}(K,F)\) the unit ball with norm \(\phi\rightarrow|\phi|^{k+2}_{(K+2)}\left(\int_{K\cap E(\phi)}|\chi,\phi|^{k+1}d\chi\right)^{\frac{-1}{(k+2)}}.\) This result holding for each subspace \(F\in G_{n,k}\) is generalized by the authors for each \textit{most} subspace. In the second part for every symmetric convex body, a bound for volume ratio of random high dimensional sections \(F\in G_{n,k}\) is given.
scientific article; zbMATH DE number 5629198
scientific article