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Computation of the ε entropy of the space of continuous functions with the Hausdorff metric

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Abstract

The number Kp,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 × p, and completely cover the side of length p of this rectangle under projection is computed. The asymptotic (Kp,q/q2)1/p → λ, as p, q → ∞, where λ = 0.3644255… is the maximum root of the equation1F1(-1/2 − 1/(16λ), 1/2, 1/(4λ)) = 0,1F1 being the confluence hypergeometric function, is established. These results allow us to compute the ε entropy of the space of continuous functions with the Hausdorff metric.

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Literature cited

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Authors and Affiliations

  1. Moscow Mining Institute, USSR

    A. A. Panov

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  1. A. A. Panov
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Translated from Matematicheskie Zametki, Vol. 21, No. 1, pp. 39–50, January, 1977.

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Panov, A.A. Computation of the ε entropy of the space of continuous functions with the Hausdorff metric. Mathematical Notes of the Academy of Sciences of the USSR 21, 22–28 (1977). https://doi.org/10.1007/BF02317030

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  • DOI: https://doi.org/10.1007/BF02317030

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