ISSN:
1420-8997
Keywords:
Primary 52.A30
;
52.A35
;
Secondary 52.A10
Source:
Springer Online Journal Archives 1860-2000
Topics:
Mathematics
Notes:
Abstract For eachk andd, 1≤k≤d, definef(d, d)=d+1 andf(d, k)=2d if 1≤k≤d−1. The following results are established: Let $$\mathcal{M}$$ be a uniformly bounded collection of compact, convex sets inR d . For a fixedk, 1≤k≤d, dim ∩{M∶M in $$\mathcal{M}$$ }≥k if and only if for someα 〉 0, everyf(d, k) members of $$\mathcal{M}$$ contain a commonk-dimensional set of measure (volume) at leastα. LetS be a bounded subset ofR d . Assume that for some fixedk, 1≤k≤d, there exists a countable family of (k−l)-flats {H i ∶:i≥1} inR d such that clS ∼S ∼ ∪{Hi ∶i ≥ 1 } and for eachi≥1, (clS ∼S) ∩H i has (k−1) dimensional measure zero. Every finite subset ofS sees viaS a set of positivek-dimensional measure if and only if for someα〉0, everyf(d,k) points ofS see viaS a set ofk-dimensional measure at least α. The numbers off(d,d) andf(d, 1) above are best possible.
Type of Medium:
Electronic Resource
URL:
http://dx.doi.org/10.1007/BF01230358
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