Metric entropy of some classes of sets (Q1363473)

Let \(Z_n\) be the set of closed star-shaped subsets with respect to the origin of the unit ball in the \(n\)-dimensional Euclidean space \(\mathbb R^n\). Let \(U_{n,k}\) be the set of closed subsets \(X\) of the \(n\)-dimensional cube \(\{x\in\mathbb R^n: |x_i|\leq 1,\;i=1,2,\dots ,n\}\) such that each \(X\in U_{n,k}\) intersects the linear span of the first \(k\) coordinate vectors in a convex set. Let \(U_{n,k}'\) be the subset of \(U_{n,k}\) such that each \(X\in U_{n,k}'\) contains the \((n-k)\)-dimensional cube \(\{x\in\mathbb R^n: x_i=0\) if \(i\leq k\) and \(|x_i|\leq 1\) if \(i>k\)\}. All the sets are endowed with the Hausdorff metric. Let \(H(K,\varepsilon)\) be the metric entropy of a compact metric space \(K\). For two functions \(f\) and \(g\) defined on the set of positive reals, the author writes \(f(\varepsilon )\asymp g(\varepsilon)\) if there exist constants \(A\) and \(B\) such that the inequalities \(f(\varepsilon)/g(\varepsilon)<A\) and \(g(\varepsilon )/f(\varepsilon)<B\) hold for \(0<\varepsilon <\varepsilon_0\). The main results of the article under review are as follows: (i) \(H(Z_n,\varepsilon)\asymp (1/\varepsilon)^{n-1} \log_2(1/\varepsilon)\); (ii) \(H(U'_{n,k},\varepsilon )\asymp (1/\varepsilon)^{n-1} \log_2(1/\varepsilon)\) if \(k<n\); (iii) \(H(U_{n,k},\varepsilon)\asymp (1/\varepsilon)^{n-1}\) if \(1<k<n\). The results (i)--(iii) are complements to the following previously known facts: \(H(U_{n,1},\varepsilon)\asymp (1/\varepsilon)^{n-1}\) [\textit{B. Penkov} and \textit{Bl. Sendov}, C. R. Acad. Bulgare Sci. 17, No.2, 335-337 (1964; Zbl 0143.35701)]; \(H(U_{n,n},\varepsilon)\asymp H(U'_{n,n},\varepsilon )\asymp (1/\varepsilon)^{(n-1)/2}\) [\textit{E. M. Bronstejn}, Sib. Math. J. 17, No.3, 393-398 (1976; Zbl 0354.54026)].