This paper relates the differential entropy of a sufficiently nice probability density function
p on Euclidean
n-space to the problem of tiling
n-space by the translates of a given compact symmetric convex set
S with nonempty interior. The relationship occurs via the concept of the epsilon entropy of
n-space under the norm induced by
S, with probability induced by
p. An expression is obtained for this entropy as
approaches 0, which equals the differential entropy of
p, plus
n times the logarithm of 2/
, plus the logarithm of the reciprocal of the volume of
S, plus a constant
C(S) depending only on
S, plus a term approaching zero with
. The constant
C(S) is called the entropic packing constant of
S; the main results of the paper concern this constant. It is shown that
C(S) is between 0 and 1; furthermore,
C(S) is zero if and only if translates of
S tile all of
n-space.This paper presents the results of one phase of research carried out at the Jet Propulsion Laboratory, California Institute of Technology, under Contract No. NAS 7-100, sponsored by the National Aeronautics and Space Administration.
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