Thick sets and quasisymmetric maps
1994 ◽
Vol 135
◽
pp. 121-148
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Keyword(s):
1.1. Thickness. Let E be a real inner product space. For a finite sequence of points a0, . . . ,ak in E we let a0. . . ,ak denote the convex hull of the set {a0, . . . , ak}. If these points are affinely independent, the set Δ = a0. . .ak is a k-simplex with vertices a0. . . ,ak. It has a well-defined k-volume written as mk(Δ) or briefly as m(Δ). We are interested in sets A ⊂ E which are “nowhere too flat in dimension k”. More precisely, suppose that A ⊂ E, q > 0 and that k: is a positive integer. We let denote the closed ball with center x and radius r. We say that A is (q, k)-thick if for each x ∈ A and r> 0 such that A\ ≠ there is a k-simplex Δ with vertices in A ∩ such that mk(Δ) ≥ qr.
1994 ◽
Vol 37
(3)
◽
pp. 338-345
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Keyword(s):
1988 ◽
Vol 30
(3)
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pp. 263-270
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2004 ◽
Vol 141
(1)
◽
pp. 1-10
◽
1976 ◽
Vol 16
(4)
◽
pp. 341-346
◽
2006 ◽
Vol 4
(1)
◽
pp. 1-6
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Keyword(s):
1996 ◽
Vol 202
(3)
◽
pp. 1040-1057
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Keyword(s):
2007 ◽
Vol 82
(3)
◽
pp. 325-344