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Название: Differential entropy and tiling
Авторы: Edward C. Posner, Eugene R. Rodemich
his paper relates the differential entropy of a sufficiently nice probability density functionp on Euclideann-space to the problem of tilingn-space by the translates of a given compact symmetric convex setS with nonempty interior. The relationship occurs via the concept of the epsilon entropy ofn-space under the norm induced byS, with probability induced byp. An expression is obtained for this entropy asapproaches 0, which equals the differential entropy ofp, plusn times the logarithm of 2/, plus the logarithm of the reciprocal of the volume ofS, plus a constantC(S) depending only onS, plus a term approaching zero with. The constantC(S) is called the entropic packing constant ofS; the main results of the paper concern this constant. It is shown thatC(S) is between 0 and 1; furthermore,C(S) is zero if and only if translates ofS tile all ofn-space.