Computation of the $\varepsilon$ entropy of the space of continuous functions with the Hausdorff metric

A. A. Panov

Matematičeskie zametki, Tome 21 (1977) no. 1, pp. 39-50

Résumé

The number $K_{p,q}$, i.e., the number of $(p,q)$ corridors of closed domains which are convex in the vertical direction, consist of elementary squares of the integral lattice, are situated within a rectangle of the size $q\times p$, and completely cover the side of length $p$ of this rectangle under projection is computed. The asymptotic $(K_{p,q}/q^2)^{1/p}\to\lambda$, as $p,q\to\infty$, where $\lambda=0,\!3644255\dots$ is the maximum root of the equation $_1F_1(-1/2-1/(16\lambda),1/2,1/(4\lambda))=0$, $_1F_1$ being the confluence hypergeometric function, is established. These results allow us to compute the $\varepsilon$ entropy of the space of continuous functions with the Hausdorff metric.