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Regularity of Distance Measures and Sets

✍ Scribed by Pertti Mattila; Per Sjölin

Publisher
John Wiley and Sons
Year
1999
Tongue
English
Weight
236 KB
Volume
204
Category
Article
ISSN
0025-584X

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✦ Synopsis

Abstract

Let μ be a Radon measure with compact support in IR^n^ such that

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We 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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