Regularity of Distance Measures and Sets | Zendy

Mattila Pertti | Zendy; Sjölin Per | Zendy

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Regularity of Distance Measures and Sets

Author(s) -

Mattila Pertti,

Sjölin Per

Publication year - 1999

Publication title -

mathematische nachrichten

Language(s) - English

Resource type - Journals

SCImago Journal Rank - 0.913

H-Index - 50

eISSN - 1522-2616

pISSN - 0025-584X

DOI - 10.1002/mana.19992040110

Subject(s) - mathematics , corollary , hausdorff dimension , hausdorff measure , measure (data warehouse) , radon measure , outer measure , hausdorff distance , combinatorics , absolute continuity , compact space , hausdorff space , dimension (graph theory) , class (philosophy) , image (mathematics) , mathematical analysis , set (abstract data type) , pure mathematics , locally compact space , minkowski–bouligand dimension , fractal dimension , fractal , artificial intelligence , database , computer science , programming language

Let μ be a Radon measure with compact support in IR n such thatWe show that the imw of μ x μ under the distance map (x, y) → |x‐ y| is an absolutely continuous measure with density of class C a ‐(n+1)/2. As a corollary we get that If AC IR n is a Suslin set with Hausdorff dimension greater than (n+1)/2, then the distance set {|x‐y| : x, y ϵ A} has non‐empty interior.

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