How Well can Space be Packed with Smooth Bodies? Measure Theoretic Results | Zendy

Gruber Peter M. | Zendy

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How Well can Space be Packed with Smooth Bodies? Measure Theoretic Results

Author(s) -

Gruber Peter M.

Publication year - 1995

Publication title -

journal of the london mathematical society

Language(s) - English

Resource type - Journals

SCImago Journal Rank - 1.441

H-Index - 62

eISSN - 1469-7750

pISSN - 0024-6107

DOI - 10.1112/jlms/52.1.1

Subject(s) - hausdorff measure , mathematics , class (philosophy) , regular polygon , bounded function , hausdorff dimension , hausdorff distance , hausdorff space , boundary (topology) , convex set , packing dimension , convex body , combinatorics , dimension (graph theory) , differentiable function , measure (data warehouse) , space (punctuation) , pure mathematics , fractal dimension , mathematical analysis , geometry , convex hull , minkowski–bouligand dimension , fractal , computer science , convex optimization , artificial intelligence , database , operating system

How well can d be packed with smooth bodies? If the bodies are convex and of class ℒ 1 , then the set not covered is quite large: it has Hausdorff dimension at least d − 2. If the bodies are convex and of class ℒ ω , that is, analytic, the set not covered has Hausdorff dimension at least d − 1. Both estimates are best possible. The situation changes drastically if non‐convex bodies are admitted: there are tilings of d with topological cells of class ℒ ∞ . In addition, for d = 2, 3 one may assume that the diameters of the tiles are uniformly bounded, but the differentiability assumptions cannot be improved substantially: for any packing of d with bodies having connected boundaries of class ℒ ω the set not covered has Hausdorff dimension at least d − 1.

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