Menger curvature and C-1 regularity of fractals

Authors
Lin, Y Mattila, P
Citation
Y. Lin et P. Mattila, Menger curvature and C-1 regularity of fractals, P AM MATH S, 129(6), 2001, pp. 1755-1762
Citations number
12
Categorie Soggetti
Mathematics
Journal title
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY
ISSN journal
00029939 → ACNP
Volume
129
Issue
6
Year of publication
2001
Pages
1755 - 1762
Database
ISI
SICI code
0002-9939(2001)129:6<1755:MCACRO>2.0.ZU;2-9
Abstract
We show that if E is an s-regular set in R-2 for which the triple integral integral (E) integral (E) integral (E) (c)(x, y, z)(2s) dH(s) x dH(s) y dH( s) z of the Menger curvature c is finite and if 0 < s <less than or equal t o> 1/2, then H-s almost all of E can be covered with countably many C-1 cur ves. We give an example to show that this is false for 1/2 < s<1.