Entropy, capacity and arrangements on the cube (Q1368003)

Let \(E\) be a real normed space and \(X \subset E\) a fixed bounded subset of \(E\). The closed balls of radius \(r>0\) centered in \(x_0\in E\) are denoted by \(B(x_0,r)\). The \(n-\)th entropy number of \(X, n\in\mathbb N\), is defined by \(\varepsilon_n(X) = \inf\{ \varepsilon > 0 :\) there exists a covering of \(X\) consisting of at most \(n\) balls of radius \(\varepsilon\}\). The \(n\)-th capacity number (or inner entropy number) of \(X\) is given by \(\varphi_n(X) = \sup\{\rho > 0 :\) there exists a \(\rho\)-distant subset of \(X\) of cardinality \(n + 1\}\) if card\((X) > n+1\), and by \(\varphi_n(X) = 0\) if card\((X) < n + 1\). The authors get the following characterization of the capacity numbers of \(X\): Let \(X\) be a convex and bounded infinite subset of a normed space \(E\). Then \(\varphi_n(X) = \sup\{ \rho > 0 :\) there exists a packing in \(X + B(0, \rho)\) consisting of \(n + 1\) balls of radius \(\rho\}\). The capacity numbers of a ball in \(E\) can be given not only as a kind of packing numbers in a parallel set as above but also as packing numbers in the ball itself: \(\varphi_n(B(x_0,r))=r\mu_n(B(x_0,r))/ (r-\mu_n(B(x_0,r)))\) with \(\mu_n(B(x_0,r)) = \sup\{\mu\in (0, r):\) there exists a packing in \( B(x_0,r)\) consisting of \(n + 1\) balls of radius \(\mu\}\). After that entropy numbers and capacity numbers of the unit ball of \(l_\infty^d\) are computed. Finally some open questions are described.