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Epsilon entropy and the packing of balls in Euclidean space
Author(s) -
Hawkes John
Publication year - 1996
Publication title -
mathematika
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.955
H-Index - 29
eISSN - 2041-7942
pISSN - 0025-5793
DOI - 10.1112/s0025579300011566
Subject(s) - mathematics , combinatorics , disjoint sets , ball (mathematics) , lebesgue measure , unit sphere , euclidean geometry , euclidean space , lebesgue integration , geometry , mathematical analysis
Summary Let{ B j = B ( x j , a j ) }j = 1 j = ∞be a sequence of mutually disjoint open balls, with centres x j and corresponding radii a j , each contained in the closed unit ballB ¯ = B ¯ ( 0 , 1 )in d ‐dimensional euclidean space, ℝ d . Further we suppose, for simplicity, that the balls B j are indexed so that a j ≥ a j +1. The set R =B ¯ / {∪ j = 1 ∞B j}obtained by removing, fromB ¯the balls { B j } is called the residual set. We say that the balls { B j } constitute a packing ofB ¯provided that λ(ℛ)=0, where λ denotes the d ‐dimensional Lebesgue measure. Thus it follows that λ ( B ¯ ) = π 1 2 d / Γ ( 1 2 d + 1 ) , henceforth denoted by c ( d ), whilst the packing restraint ensures that∑ j = 1 ∞a j d = 1 . Larman [11] has noted that, under these circumstances, one also has∑ j = 1 ∞a j d − 1 = ∞.